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"Off-The-Grid" Methods

"Off-The-Grid" Methods

Yohann De Castro

June 26, 2018
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  1. Yohann DE CASTRO THREE CONES Orthant x 0 <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit>

    <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> Vectors • High-Dimensional Statistics
  2. • Matrix completion, recovery, factorisation Yohann DE CASTRO THREE CONES

    Orthant SDP x 0 <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> M ⌫ 0 <latexit sha1_base64="va6iNbYiR9AaqO+CfS1XfAKZo0c=">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</latexit> <latexit sha1_base64="va6iNbYiR9AaqO+CfS1XfAKZo0c=">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</latexit> <latexit sha1_base64="va6iNbYiR9AaqO+CfS1XfAKZo0c=">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</latexit> <latexit sha1_base64="va6iNbYiR9AaqO+CfS1XfAKZo0c=">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</latexit> Vectors Matrices • High-Dimensional Statistics
  3. • Global Optimization • Sparse Deconvolution • Computational Limits Yohann

    DE CASTRO THREE CONES Orthant SDP Lasserre’s hierarchies x 0 <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> M ⌫ 0 <latexit 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  4. • Global Optimization • Sparse Deconvolution • Computational Limits Yohann

    DE CASTRO THREE CONES Orthant SDP Lasserre’s hierarchies x 0 <latexit sha1_base64="khPW6/AJ8c0qeirP+HxMftEEt6w=">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</latexit> <latexit 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  5. Yohann DE CASTRO Sparse Deconvolution changing the rates of photoactivation

    and photobleaching, even if the density of fluorophore-labeled molecules is much higher than one fluorophore per mm2. We present a novel method by which fluorescence micros- copy may be performed to obtain an image with greatly enhanced ability to resolve large numbers of fluorescent 1. The spontaneous interconversion activated (fluorescent) state mus light-controlled activation rate. 2. For irreversible photoactivatio quantum yield must be finite a FIGURE 1 localization area contain (here, PA- neously wit for readout spatial illum a second on diode laser, Within the vation beam blue circles circles) and time, the ac (red Xs) an (black circl then activated, localized, and bleached until a sufficient number of molecules have been analyzed to construct an image. (G) Th the 405-nm activation laser (X405), which is reflected by a dichroic (DM1) to make it collinear with the Ar1 readout laser. A inverted fluorescence microscope is used to focus the lasers, which are reflected upward by a second dichroic mirror (DM2 objective lens (OBJ). The sample, supported by a coverslip (CS), emits fluorescence which is collected by the objective, transm and focused by the tube lens (TL) to form an image on a camera (CCD). Biophysica S. Hess, T. Girirajan, M. Mason, Ultra-High Resolution Imaging by Fluorescence Photoactivation Localization Microscopy, Biophysical Journal (2004). SUPER RESOLUTION
  6. Yohann DE CASTRO Sparse Deconvolution Y. Li, S. Osher, R.

    Tsai, Heat Source Identification based on L1 Constrained Minimization, Inverse Problems and Imaging (2014). 210 Yingying Li, Stanley Osher and Richard Tsai (a) Heat source u0 (b) Au0 = f (c) f + noise SUPER RESOLUTION
  7. Yohann DE CASTRO Sparse Deconvolution H. Pan, T. Blu, M.

    Vetterli, Towards Generalized FRI Sampling With an Application to Source Resolution in Radioastronomy, IEEE trans. on Signal Processing (2017). EE, Thierry Blu, Fellow, IEEE, and Martin Vetterli, Fellow, IEEE continuous-time arrival problem, rliest occurrence Prony’s method. n high resolution -time sparse sig- sampling theory ectral estimation pling. But not all fied by a concrete s, we develop the , typically sum of his by identifying uniform samples uniform samples A valid solution ization such that en measurements Fig. 1. Schematic diagram of a radio interferometer. The cross-corre of the received signals at different antennas are related to the Fourier tra of the sky image (see Table I) at certain non-uniform frequencies. SUPER RESOLUTION
  8. Yohann DE CASTRO CONIC PROGRAMMING IN ML, THE BIG PICTURE

    Sparse Deconvolution x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit>
  9. Yohann DE CASTRO CONIC PROGRAMMING IN ML, THE BIG PICTURE

    Sparse Deconvolution x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit 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  10. Yohann DE CASTRO CONIC PROGRAMMING IN ML, THE BIG PICTURE

    Sparse Deconvolution x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit 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sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> y <latexit 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sha1_base64="Ne219FMGE9f9p0Jycnzrb+NuVNQ=">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</latexit> <latexit sha1_base64="Ne219FMGE9f9p0Jycnzrb+NuVNQ=">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</latexit> <latexit sha1_base64="Ne219FMGE9f9p0Jycnzrb+NuVNQ=">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</latexit> Perfect Recovery y = (x) <latexit sha1_base64="gYqwh+jPG1WNxxB30eg3MeuJco4=">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</latexit> <latexit sha1_base64="gYqwh+jPG1WNxxB30eg3MeuJco4=">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</latexit> <latexit sha1_base64="gYqwh+jPG1WNxxB30eg3MeuJco4=">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</latexit> <latexit sha1_base64="gYqwh+jPG1WNxxB30eg3MeuJco4=">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</latexit> min y= (x) ||x||TV <latexit 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sha1_base64="tFNtusX5sUh/CrZYwZLoy8tctKQ=">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</latexit> <latexit 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sha1_base64="tFNtusX5sUh/CrZYwZLoy8tctKQ=">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</latexit> ||x||TV = min x=x+ x ,x± 0 Z dx+ + Z dx <latexit 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sha1_base64="p4mRkdaDz8YflqGHACejV7QE9K4=">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</latexit> <latexit sha1_base64="p4mRkdaDz8YflqGHACejV7QE9K4=">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</latexit>
  11. Yohann DE CASTRO CONIC PROGRAMMING IN ML, THE BIG PICTURE

    Sparse Deconvolution x <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> <latexit sha1_base64="MyiEmFnntf+qNJvnSMVFXu6341g=">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</latexit> y <latexit sha1_base64="Ne219FMGE9f9p0Jycnzrb+NuVNQ=">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</latexit> <latexit sha1_base64="Ne219FMGE9f9p0Jycnzrb+NuVNQ=">AAACxHicjVHLSsNAFD2Nr1pfVZdugkVwVRIRdFkUxGUL9gG1SJJO69A0CTMToRT9Abf6beIf6F94Z5yCWkQnJDlz7j1n5t4bZjGXyvNeC87C4tLySnG1tLa+sblV3t5pyTQXEWtGaZyKThhIFvOENRVXMetkggXjMGbtcHSu4+07JiRPkys1yVhvHAwTPuBRoIhqTG7KFa/qmeXOA9+CCuyqp+UXXKOPFBFyjMGQQBGOEUDS04UPDxlxPUyJE4S4iTPco0TanLIYZQTEjug7pF3Xsgnttac06ohOiekVpHRxQJqU8gRhfZpr4rlx1uxv3lPjqe82oX9ovcbEKtwS+5dulvlfna5FYYBTUwOnmjLD6Ooi65Kbruibu1+qUuSQEadxn+KCcGSUsz67RiNN7bq3gYm/mUzN6n1kc3O861vSgP2f45wHraOq71X9xnGldmZHXcQe9nFI8zxBDZeoo2m8H/GEZ+fCiR3p5J+pTsFqdvFtOQ8fbXyPfg==</latexit> <latexit sha1_base64="Ne219FMGE9f9p0Jycnzrb+NuVNQ=">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</latexit> <latexit sha1_base64="Ne219FMGE9f9p0Jycnzrb+NuVNQ=">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</latexit> Perfect Recovery y = (x) <latexit sha1_base64="gYqwh+jPG1WNxxB30eg3MeuJco4=">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</latexit> <latexit sha1_base64="gYqwh+jPG1WNxxB30eg3MeuJco4=">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</latexit> <latexit 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sha1_base64="gYqwh+jPG1WNxxB30eg3MeuJco4=">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</latexit> Discrete Measure min y= (x) ||x||TV <latexit 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  12. Yohann DE CASTRO CONIC PROGRAMMING IN ML, THE BIG PICTURE

    Sparse Deconvolution = min y= (x+) (x ) , x± 0 h1, x+ i + h1, x i <latexit sha1_base64="XRjxFqIFSX3iKrbjDnuZ439iDhI=">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</latexit> <latexit 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  13. [Cone] Moments of Nonnegative Measures
 [Algorithms] Lasserre’s Hierarchies, Frank-Wolfe
 Yohann

    DE CASTRO CONIC PROGRAMMING IN ML, THE BIG PICTURE Sparse Deconvolution = min y= (x+) (x ) , x± 0 h1, x+ i + h1, x i <latexit sha1_base64="XRjxFqIFSX3iKrbjDnuZ439iDhI=">AAADHnicjVHLThRBFD20Iogioy7dVJyYQJiZdBMS3JAQ3LgcE4dHKNLpboqhMtUPqqsNkwn/4p+wc2dYoh9gohv9BW8VRSIQgtXp7nPPvedU3bpppWRtwvD7VPDg4fSjmdnHc0+ezj9baD1/sVWXjc7EICtVqXfSpBZKFmJgpFFip9IiyVMlttPRO5vf/iR0LcvioxlXYj9PhoU8lFliiIpbu+s8l0U8Ga/z/pFcPImXl7oedZd4p8PZScyrnA/FMQtPGVdJMVSCRR2q5PoyWL5Gdz0dt9phL3SL3QaRB2341S9bF+A4QIkMDXIIFDCEFRLU9OwhQoiKuH1MiNOEpMsLnGKOtA1VCapIiB3Rd0jRnmcLiq1n7dQZ7aLo1aRkeEOakuo0Ybsbc/nGOVv2Lu+J87RnG9M/9V45sQZHxN6nu6r8X53txeAQb10PknqqHGO7y7xL427Fnpz905Uhh4o4iw8orwlnTnl1z8xpate7vdvE5X+6SsvaOPO1DX7ZU9KAo5vjvA22VnpR2Is+rLY3Nv2oZ/EKr7FI81zDBt6jjwF5n+EHfuNP8Dn4EnwNzi9LgymveYlrK/j2F6hvsFs=</latexit> <latexit 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  14. [Cone] Moments of Nonnegative Measures
 [Algorithms] Lasserre’s Hierarchies, Frank-Wolfe
 Yohann

    DE CASTRO CONIC PROGRAMMING IN ML, THE BIG PICTURE Sparse Deconvolution Linear Conic with « Moments » Constraints = min y= (x+) (x ) , x± 0 h1, x+ i + h1, x i <latexit sha1_base64="XRjxFqIFSX3iKrbjDnuZ439iDhI=">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</latexit> <latexit 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sha1_base64="mpmpiO8OlfPxt0dlsLYovjTwngo=">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</latexit>
  15. Yohann DE CASTRO Robust Moment Problem : Robustness SUPER RESOLUTION

    ¯ x 2 arg min v n1 2 ||y (v)||2 2 + ||v||TV o <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit 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  16. Yohann DE CASTRO Robust Moment Problem [DC et al; Candès

    and Fernandez-Granda; Recht; …] Spike Detection : Robustness SUPER RESOLUTION ¯ x 2 arg min v n1 2 ||y (v)||2 2 + ||v||TV o <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit 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sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> y = (x) + ⌘ <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> • Analysis via « Dual Certificate » [DC, Gamboa; Candès, Tao; …] • « Minimal Separation Condition » on the support of the spikes • Optimal error bounds • [DC, Gamboa, Henrion, Lasserre] 
 Tractable Algorithm via « Lasserre’s hierarchies »
  17. Yohann DE CASTRO Robust Moment Problem : Robustness SUPER RESOLUTION

    ¯ x 2 arg min v n1 2 ||y (v)||2 2 + ||v||TV o <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> y = (x) + ⌘ <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> • Analysis via « Dual Certificate » [DC, Gamboa; Candès, Tao; …] • « Minimal Separation Condition » on the support of the spikes • Optimal error bounds • [DC, Gamboa, Henrion, Lasserre] 
 Tractable Algorithm via « Lasserre’s hierarchies » + Support Recovery [DC et al; Candès and Fernandez-Granda; Recht; …] Spike Detection
  18. Yohann DE CASTRO Robust Moment Problem : Robustness SUPER RESOLUTION

    ¯ x 2 arg min v n1 2 ||y (v)||2 2 + ||v||TV o <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">AAADHXicjVHLbhMxFD0dXqW8AizZWERIRYhoJkKCZdVuWAapSSPVJfI4TmJ1XvJ4IqKZfAt/0l13FduKPQtYwDdw7U4loELg0cwcn3vPse+9cZHo0obh543g2vUbN29t3t66c/fe/Qedh49GZV4ZqYYyT3IzjkWpEp2podU2UePCKJHGiTqIj/dc/GCpTKnzbN+uCnWUinmmZ1oKS9SkM+axMOwD1xkXZs5TnU3q5Zrv6jmv+XRmhIzq/rppVi/5YKG3l8+bZtJ/32cvGE/olKlomiVRNU+FXZiU7Y8uxOtJpxv2Qr/YVRC1oIt2DfLOOTimyCFRIYVCBks4gUBJzyEihCiIO0JNnCGkfVxhjS3SVpSlKEMQe0zfOe0OWzajvfMsvVrSKQm9hpQMz0iTU54h7E5jPl55Z8f+zbv2nu5uK/rHrVdKrMWC2H/pLjP/V+dqsZjhja9BU02FZ1x1snWpfFfczdkvVVlyKIhzeEpxQ1h65WWfmdeUvnbXW+HjX32mY91etrkVvrlb0oCjP8d5FYz6vSjsRe9edXd221Fv4gmeYpvm+Ro7eIsBhuR9gi/4jh/Bx+A0OAs+XaQGG63mMX5bwflPXh+zzQ==</latexit> <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> y = (x) + ⌘ <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> • Analysis via « Dual Certificate » [DC, Gamboa; Candès, Tao; …] • « Minimal Separation Condition » on the support of the spikes • Optimal error bounds • [DC, Gamboa, Henrion, Lasserre] 
 Tractable Algorithm via « Lasserre’s hierarchies » + Support Recovery [DC et al; Candès and Fernandez-Granda; Recht; …] Spike Detection
  19. Yohann DE CASTRO Robust Moment Problem [DC; Candès and Fernandez-Granda;

    Recht; …] Spike Detection : Robustness SUPER RESOLUTION ¯ x 2 arg min v n1 2 ||y (v)||2 2 + ||v||TV o <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> <latexit sha1_base64="PGoF5aJAATg57JT9Bs8iAqC7Q+Q=">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</latexit> y = (x) + ⌘ <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> <latexit sha1_base64="wN9oEnJY+4YijXouSL/KYNrpMaY=">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</latexit> • Analysis via « Dual Certificate » [DC, Gamboa; Candès, Tao; …] • « Minimal Separation Condition » on the support of the spikes • Optimal error bounds • [DC, Gamboa, Henrion, Lasserre] 
 Tractable Algorithm via « Lasserre’s hierarchies » + Support Recovery Estimated signal with Blasso Original signal Constant Number of Moments <latexit sha1_base64="tjPHPXadtt6oxYHWW323bfrlZc0=">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</latexit> <latexit sha1_base64="tjPHPXadtt6oxYHWW323bfrlZc0=">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</latexit> <latexit sha1_base64="tjPHPXadtt6oxYHWW323bfrlZc0=">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</latexit> <latexit sha1_base64="tjPHPXadtt6oxYHWW323bfrlZc0=">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</latexit>
  20. thin grids Grid-less approach Grid approach Inference from grid points

    Inference from the continuous process p Yohann DE CASTRO STATISTICAL INFERENCE LIMITS Grid vs Off-the-grid [ADCM18]
  21. thin grids Grid-less approach Grid approach Inference from grid points

    p Yohann DE CASTRO STATISTICAL INFERENCE LIMITS Grid vs Off-the-grid [ADCM18] Inference from the continuous process
  22. thin grids Grid-less approach Grid approach Inference from grid points

    Inference from the process X(·) p Yohann DE CASTRO STATISTICAL INFERENCE LIMITS Grid vs Off-the-grid [ADCM18] Under one Sparse alternative Grids Off-the-grid
  23. Yohann DE CASTRO LASSERRE HIERARCHIES f(t) <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">AAACx3icjVHLTsJAFD3UF+ILdemmkZjgpmkLKOxI3OgOE0ESJKYtAzT0lXZKJMSFP+BW/8z4B/oX3hlLogui07S9c+45Z+bea0eem3Bdf88pK6tr6xv5zcLW9s7uXnH/oJOEaeywthN6Ydy1rYR5bsDa3OUe60Yxs3zbY7f25ELkb6csTtwwuOGziPV9axS4Q9exuICGZX56XyzpWt2sNOo1Vddqptk4q4pAb1SMqmpoulwlZKsVFt9whwFCOEjhgyEAp9iDhYSeHgzoiAjrY05YTJEr8wyPKJA2JRYjhkXohL4j2vUyNKC98Eyk2qFTPHpjUqo4IU1IvJhicZoq86l0Fugy77n0FHeb0d/OvHxCOcaE/qVbMP+rE7VwDFGXNbhUUyQRUZ2TuaSyK+Lm6o+qODlEhIl4QPmYYkcqF31WpSaRtYveWjL/IZkCFXsn46b4FLekAS+mqC4POqZm6JpxXS01m9mo8zjCMco0z3M0cYkW2uQ9xjNe8KpcKaEyVR6+qUou0xzi11KevgBLiZCh</latexit> <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit>

    <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit> <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit>
  24. Yohann DE CASTRO LASSERRE HIERARCHIES Newton’s approach f(t) <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit>

    <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit> <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit> <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit> Search through local optima Well suited for convex functions BUT simple non-convex function
 may have an exponential number
 of bad optima, e.g. f(t1) + . . . + f(tn) <latexit sha1_base64="rBPCHuh1dvDMS67EIrT6CEjCSRI=">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</latexit> <latexit sha1_base64="rBPCHuh1dvDMS67EIrT6CEjCSRI=">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</latexit> <latexit sha1_base64="rBPCHuh1dvDMS67EIrT6CEjCSRI=">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</latexit> <latexit sha1_base64="rBPCHuh1dvDMS67EIrT6CEjCSRI=">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</latexit>
  25. Yohann DE CASTRO LASSERRE HIERARCHIES Newton’s approach Hilbert’s approach 3

    2 (t 1)2 8 (t2 1)2 <latexit sha1_base64="PL1HPibmFkFrEK63EhsulkoJqY4=">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</latexit> <latexit sha1_base64="PL1HPibmFkFrEK63EhsulkoJqY4=">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</latexit> <latexit sha1_base64="PL1HPibmFkFrEK63EhsulkoJqY4=">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</latexit> <latexit sha1_base64="PL1HPibmFkFrEK63EhsulkoJqY4=">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</latexit> f(t) = <latexit sha1_base64="5QPmFxcaii2LWoNYAIW+1bM5KVw=">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</latexit> <latexit sha1_base64="5QPmFxcaii2LWoNYAIW+1bM5KVw=">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</latexit> <latexit sha1_base64="5QPmFxcaii2LWoNYAIW+1bM5KVw=">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</latexit> <latexit sha1_base64="5QPmFxcaii2LWoNYAIW+1bM5KVw=">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</latexit> f(t) <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit> <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit> <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit> <latexit sha1_base64="KKfdZdAuoCwUfI847X7NtAkgDvM=">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</latexit> Search through local optima Well suited for convex functions BUT simple non-convex function
 may have an exponential number
 of bad optima, e.g. f(t1) + . . . + f(tn) <latexit sha1_base64="rBPCHuh1dvDMS67EIrT6CEjCSRI=">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</latexit> <latexit sha1_base64="rBPCHuh1dvDMS67EIrT6CEjCSRI=">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</latexit> <latexit sha1_base64="rBPCHuh1dvDMS67EIrT6CEjCSRI=">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</latexit> <latexit sha1_base64="rBPCHuh1dvDMS67EIrT6CEjCSRI=">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</latexit> Globally decompose the function
 into simple pieces Lasserre’s hierarchy algorithm: 
 find decomposition efficiently Strongest provable guarantees
 for LOTS of ML problems
  26. Yohann DE CASTRO Global Optimization LASSERRE HIERARCHIES min t2⌦ f(t)

    = min µ 0 , µ(⌦)=1 Z ⌦ f(t)µ(dt) <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit> <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">AAADHXicjVFNT9tAEH0YWj76QVqOXKxGSIkURTZCKpdIqL30BkgNRIqRtXY2YYXXdtfrSpGV39J/wo0b6hVx59Ae4DcwuzESLarKWrbfvHlvdmcnyhNRaM+7XnAWl168XF5ZXXv1+s3b9ca790dFVqqY9+MsydQgYgVPRMr7WuiED3LFmYwSfhydfTb54+9cFSJLv+ppzk8km6RiLGKmiQobg0CKNKx0INJgX/IJm41but2bs4Esgwn/5npBpxO4FLXmmnbPn5FBh/PQNRablUyfKlmNZrodNppe17PLfQr8GjRRr4OscYUAI2SIUUKCI4UmnIChoGcIHx5y4k5QEacICZvnmGGNvCWpOCkYsWf0nVA0rNmUYlOzsO6YdknoVeR0sUWejHSKsNnNtfnSVjbsv2pXtqY525T+UV1LEqtxSuz/fA/K5/pMLxpj7NoeBPWUW8Z0F9dVSnsr5uTuo640VciJM3hEeUU4ts6He3atp7C9m7tlNv/LKg1r4rjWlvhtTkkD9v8e51NwtN31va5/uNPc+1SPegWb+IAWzfMj9vAFB+hT7XPc4BZ3zg/nwrl0fs6lzkLt2cAfy7m6B55NslE=</latexit> <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">AAADHXicjVFNT9tAEH0YWj76QVqOXKxGSIkURTZCKpdIqL30BkgNRIqRtXY2YYXXdtfrSpGV39J/wo0b6hVx59Ae4DcwuzESLarKWrbfvHlvdmcnyhNRaM+7XnAWl168XF5ZXXv1+s3b9ca790dFVqqY9+MsydQgYgVPRMr7WuiED3LFmYwSfhydfTb54+9cFSJLv+ppzk8km6RiLGKmiQobg0CKNKx0INJgX/IJm41but2bs4Esgwn/5npBpxO4FLXmmnbPn5FBh/PQNRablUyfKlmNZrodNppe17PLfQr8GjRRr4OscYUAI2SIUUKCI4UmnIChoGcIHx5y4k5QEacICZvnmGGNvCWpOCkYsWf0nVA0rNmUYlOzsO6YdknoVeR0sUWejHSKsNnNtfnSVjbsv2pXtqY525T+UV1LEqtxSuz/fA/K5/pMLxpj7NoeBPWUW8Z0F9dVSnsr5uTuo640VciJM3hEeUU4ts6He3atp7C9m7tlNv/LKg1r4rjWlvhtTkkD9v8e51NwtN31va5/uNPc+1SPegWb+IAWzfMj9vAFB+hT7XPc4BZ3zg/nwrl0fs6lzkLt2cAfy7m6B55NslE=</latexit> <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit>
  27. Yohann DE CASTRO Global Optimization LASSERRE HIERARCHIES min t2⌦ f(t)

    = min µ 0 , µ(⌦)=1 Z ⌦ f(t)µ(dt) <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit> <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit> <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">AAADHXicjVFNT9tAEH0YWj76QVqOXKxGSIkURTZCKpdIqL30BkgNRIqRtXY2YYXXdtfrSpGV39J/wo0b6hVx59Ae4DcwuzESLarKWrbfvHlvdmcnyhNRaM+7XnAWl168XF5ZXXv1+s3b9ca790dFVqqY9+MsydQgYgVPRMr7WuiED3LFmYwSfhydfTb54+9cFSJLv+ppzk8km6RiLGKmiQobg0CKNKx0INJgX/IJm41but2bs4Esgwn/5npBpxO4FLXmmnbPn5FBh/PQNRablUyfKlmNZrodNppe17PLfQr8GjRRr4OscYUAI2SIUUKCI4UmnIChoGcIHx5y4k5QEacICZvnmGGNvCWpOCkYsWf0nVA0rNmUYlOzsO6YdknoVeR0sUWejHSKsNnNtfnSVjbsv2pXtqY525T+UV1LEqtxSuz/fA/K5/pMLxpj7NoeBPWUW8Z0F9dVSnsr5uTuo640VciJM3hEeUU4ts6He3atp7C9m7tlNv/LKg1r4rjWlvhtTkkD9v8e51NwtN31va5/uNPc+1SPegWb+IAWzfMj9vAFB+hT7XPc4BZ3zg/nwrl0fs6lzkLt2cAfy7m6B55NslE=</latexit> <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit> Framework f(t) = X f↵t↵ <latexit sha1_base64="ePh/TGb5PPdv2JrAbBu4fj/znZU=">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</latexit> <latexit sha1_base64="ePh/TGb5PPdv2JrAbBu4fj/znZU=">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</latexit> <latexit sha1_base64="ePh/TGb5PPdv2JrAbBu4fj/znZU=">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</latexit> <latexit sha1_base64="ePh/TGb5PPdv2JrAbBu4fj/znZU=">AAAC4HicjVHLSsNAFD2Nr1pfVZfdBItQNyURQTdCwY3LCvYBbS2TdNqG5kUyEUpx4c6duPUH3OrXiH+gf+GdaQpqEZ2Q5My595yZe68Vuk4sDOMtoy0sLi2vZFdza+sbm1v57Z16HCSRzWt24AZR02Ixdx2f14QjXN4MI848y+UNa3Qm441rHsVO4F+Kccg7Hhv4Tt+xmSCqmy/0S+LgtB0nnt7vtpkbDpkurqagmy8aZUMtfR6YKSgiXdUg/4o2eghgI4EHDh+CsAuGmJ4WTBgIietgQlxEyFFxjhvkSJtQFqcMRuyIvgPatVLWp730jJXaplNceiNS6tgnTUB5EWF5mq7iiXKW7G/eE+Up7zamv5V6ecQKDIn9SzfL/K9O1iLQx4mqwaGaQsXI6uzUJVFdkTfXv1QlyCEkTuIexSPCtlLO+qwrTaxql71lKv6uMiUr93aam+BD3pIGbP4c5zyoH5ZNo2xeHBUrlXTUWRSwhxLN8xgVnKOKGnnf4gnPeNEs7U671x6mqVom1ezi29IePwE++5ns</latexit> ⌦ = n t 2 RD : X g↵,mt↵ 0 , 1  m  M o <latexit 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sha1_base64="+HV5kPFLONIKpQbz6++mloY5Vew=">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</latexit> <latexit sha1_base64="+HV5kPFLONIKpQbz6++mloY5Vew=">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</latexit> Polynomial Semi-Algebraic
  28. Yohann DE CASTRO Global Optimization LASSERRE HIERARCHIES min t2⌦ f(t)

    = min µ 0 , µ(⌦)=1 Z ⌦ f(t)µ(dt) <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit> <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit> <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit> <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit> Framework f(t) = X f↵t↵ <latexit sha1_base64="ePh/TGb5PPdv2JrAbBu4fj/znZU=">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</latexit> <latexit sha1_base64="ePh/TGb5PPdv2JrAbBu4fj/znZU=">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</latexit> <latexit sha1_base64="ePh/TGb5PPdv2JrAbBu4fj/znZU=">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</latexit> <latexit sha1_base64="ePh/TGb5PPdv2JrAbBu4fj/znZU=">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</latexit> ⌦ = n t 2 RD : X g↵,mt↵ 0 , 1  m  M o <latexit 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  29. Yohann DE CASTRO Global Optimization LASSERRE HIERARCHIES min t2⌦ f(t)

    = min µ 0 , µ(⌦)=1 Z ⌦ f(t)µ(dt) <latexit sha1_base64="t8NZAV8MjesFJPrfSQOKoCZQoLk=">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</latexit> <latexit 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  30. Yohann DE CASTRO LASSERRE HIERARCHIES min t2⌦ X f↵t↵ =

    min µ 0 , µ(⌦)=1 X f↵ Z ⌦ t↵µ(dt) = min (m↵)2M X f↵m↵ <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">AAADeHicjVHdbtMwGP3S8LOVvwJXiBuLCtFKU5VMSHAzUYmbcYEYEt0mzSNyXLe1ZifBcZCmqk+1h5l4A3gFrvbZcRF0QuAoyfH5zjn2Z+eVkrVNkm9RJ75x89btre3unbv37j/oPXx0WJeN4WLCS1Wa45zVQslCTKy0ShxXRjCdK3GUn7119aOvwtSyLD7Z80qcajYv5ExyZpHKehdUyyJbWioL+kGLOVvRutFkllGmqgUj9nMLunutkOqGzsUXktCdHUpwNmhtw710w4mJNmuLv1K8XjO7MHo5XdnhOnWgg2vo9uEEnCnyfiNxLcp6/WSU+EGugzSAPoRxUPYugcIUSuDQgAYBBVjEChjU+JxACglUyJ3CEjmDSPq6gBV00dugSqCCIXuG3znOTgJb4Nxl1t7NcRWFr0EngefoKVFnELvViK83Ptmxf8te+ky3t3P85yFLI2thgey/fGvl//pcLxZm8Nr3ILGnyjOuOx5SGn8qbufkt64sJlTIOTzFukHMvXN9zsR7at+7O1vm69+90rFuzoO2gR9ul3jB6eZ1XgeHu6M0GaUfX/bH43DVW/AUnsEA7/MVjGEfDmACPHoSvYn2o3ednzGJX8TDVtqJgucx/DHi3SvqONAK</latexit>
  31. Yohann DE CASTRO LASSERRE HIERARCHIES M <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit> <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit>

    <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit> <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit> M = lim inf d MSOS(d) ✓ . . . ✓ MSOS(d) ✓ MSOS(d 2) ✓ . . . ✓ MSOS(2) <latexit sha1_base64="sdPHALlfSy+laZUymUEryDRBrCg=">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</latexit> <latexit sha1_base64="sdPHALlfSy+laZUymUEryDRBrCg=">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</latexit> <latexit sha1_base64="sdPHALlfSy+laZUymUEryDRBrCg=">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</latexit> <latexit sha1_base64="sdPHALlfSy+laZUymUEryDRBrCg=">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</latexit> min t2⌦ X f↵t↵ = min µ 0 , µ(⌦)=1 X f↵ Z ⌦ t↵µ(dt) = min (m↵)2M X f↵m↵ <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit>
  32. Yohann DE CASTRO LASSERRE HIERARCHIES M <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit> <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit>

    <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit> <latexit sha1_base64="KBoRXA49gmW5btfdx71U7++O/Yc=">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</latexit> M = lim inf d MSOS(d) ✓ . . . ✓ MSOS(d) ✓ MSOS(d 2) ✓ . . . ✓ MSOS(2) <latexit sha1_base64="sdPHALlfSy+laZUymUEryDRBrCg=">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</latexit> <latexit sha1_base64="sdPHALlfSy+laZUymUEryDRBrCg=">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</latexit> <latexit sha1_base64="sdPHALlfSy+laZUymUEryDRBrCg=">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</latexit> <latexit sha1_base64="sdPHALlfSy+laZUymUEryDRBrCg=">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</latexit> Lasserre’s hierarchies give a sequence of nested OUTER
 SDP approximations
 of the cone of moments of nonnegative measures min t2⌦ X f↵t↵ = min µ 0 , µ(⌦)=1 X f↵ Z ⌦ t↵µ(dt) = min (m↵)2M X f↵m↵ <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">AAADeHicjVHdbtMwGP3S8LOVvwJXiBuLCtFKU5VMSHAzUYmbcYEYEt0mzSNyXLe1ZifBcZCmqk+1h5l4A3gFrvbZcRF0QuAoyfH5zjn2Z+eVkrVNkm9RJ75x89btre3unbv37j/oPXx0WJeN4WLCS1Wa45zVQslCTKy0ShxXRjCdK3GUn7119aOvwtSyLD7Z80qcajYv5ExyZpHKehdUyyJbWioL+kGLOVvRutFkllGmqgUj9nMLunutkOqGzsUXktCdHUpwNmhtw710w4mJNmuLv1K8XjO7MHo5XdnhOnWgg2vo9uEEnCnyfiNxLcp6/WSU+EGugzSAPoRxUPYugcIUSuDQgAYBBVjEChjU+JxACglUyJ3CEjmDSPq6gBV00dugSqCCIXuG3znOTgJb4Nxl1t7NcRWFr0EngefoKVFnELvViK83Ptmxf8te+ky3t3P85yFLI2thgey/fGvl//pcLxZm8Nr3ILGnyjOuOx5SGn8qbufkt64sJlTIOTzFukHMvXN9zsR7at+7O1vm69+90rFuzoO2gR9ul3jB6eZ1XgeHu6M0GaUfX/bH43DVW/AUnsEA7/MVjGEfDmACPHoSvYn2o3ednzGJX8TDVtqJgucx/DHi3SvqONAK</latexit> <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> <latexit sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> SDPs Non SDP
  33. Yohann DE CASTRO KHOT’S CONJECTURE Optimality of Lasserre’s hierarchies? Under

    Khot’s Unique Game Conjecture: NP-hard to achieve better approximation guarantees than Lasserre’s hierarchies for class of worst-case discrete optimization problems [Khot’02, Raghavendra’08] • Max Cut problem • Stable Set problem • Planted Clique problem • Deg. 4 on the Sphere • Tensor PCA • LOTS of problems
  34. Yohann DE CASTRO KHOT’S CONJECTURE Optimality of Lasserre’s hierarchies? •

    Max Cut problem • Stable Set problem • Planted Clique problem • Deg. 4 on the Sphere • Tensor PCA • LOTS of problems Under Khot’s Unique Game Conjecture: NP-hard to achieve better approximation guarantees than Lasserre’s hierarchies for class of worst-case discrete optimization problems [Khot’02, Raghavendra’08]
  35. Yohann DE CASTRO KHOT’S CONJECTURE Optimality of Lasserre’s hierarchies? •

    Max Cut problem • Stable Set problem • Planted Clique problem • Deg. 4 on the Sphere • Tensor PCA • LOTS of problems In Theoretical Computer Science
 
 Lasserre = Computational Limit Under Khot’s Unique Game Conjecture: NP-hard to achieve better approximation guarantees than Lasserre’s hierarchies for class of worst-case discrete optimization problems [Khot’02, Raghavendra’08]
  36. ML Applications LASSERRE HIERARCHIES -5 0.8 -4 0.6 0.4 -3

    x 2 0.2 -2 0.5 0 x 1 -0.2 0 -1 -0.4 -0.5 0 Combinatorial Optimization on Graphs Tensor Decompositions Sparse Deconvolution [DCGHL16, ADCG14, DCG13] Gaussian Mixtures [DCGMM18+] Optimal Designs [DCGHHL18] Yohann DE CASTRO
  37. CONVEX DUALITY Yohann DE CASTRO Convex Objective inf (m↵)2M s

    1 X ↵=0 f↵m↵ <latexit sha1_base64="gMjn+pzsyf317jyjeOcE7Dkj0g0=">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</latexit> <latexit 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sha1_base64="0UGu2/bvxwu5Z1zuLOLCgWfQP0c=">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</latexit> <latexit 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sha1_base64="TqvxfdxsVJ3rySWZ9KaD/fwGTFk=">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</latexit> <latexit sha1_base64="TqvxfdxsVJ3rySWZ9KaD/fwGTFk=">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</latexit> <latexit sha1_base64="TqvxfdxsVJ3rySWZ9KaD/fwGTFk=">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</latexit>
  38. CONVEX DUALITY Yohann DE CASTRO Convex Objective inf (m↵)2M s

    1 X ↵=0 f↵m↵ <latexit sha1_base64="gMjn+pzsyf317jyjeOcE7Dkj0g0=">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</latexit> <latexit sha1_base64="gMjn+pzsyf317jyjeOcE7Dkj0g0=">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</latexit> <latexit sha1_base64="gMjn+pzsyf317jyjeOcE7Dkj0g0=">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</latexit> <latexit sha1_base64="gMjn+pzsyf317jyjeOcE7Dkj0g0=">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</latexit> inf (m↵)2M F( s 1 X ↵=0 m↵H↵) <latexit sha1_base64="0UGu2/bvxwu5Z1zuLOLCgWfQP0c=">AAADDXicjVFNSxxBEH2OGo2JcTXHXAaXwObgMiOCXoSFgOxFUHBVcMzQ0/a6jT0fTPcIMuxvyD/x5k1yzR8QQg7xnPwLq9tZSCLB9DAzr17Ve93VlRRKahME36e86ZnZF3PzLxdevV58s9RaXjnUeVVyMeC5ysvjhGmhZCYGRholjotSsDRR4ii5+GjzR5ei1DLPDsxVIU5Tdp7JoeTMEBW3+pHMhnHdSeOIqWLEPlAcpcyMOFP+7ninE+kqjevH5HYw/lTrtXDsT8r9/kQXt9pBN3DLfwrCBrTRrL289Q0RzpCDo0IKgQyGsAKDpucEIQIUxJ2iJq4kJF1eYIwF0lZUJaiCEXtB33OKTho2o9h6aqfmtIuitySlj/ekyamuJGx3812+cs6W/Zd37Tzt2a7onzReKbEGI2Kf000q/1dnezEYYsv1IKmnwjG2O964VO5W7Mn937oy5FAQZ/EZ5UvC3Ckn9+w7jXa927tlLv/TVVrWxryprfDLnpIGHP49zqfgcL0bBt1wf6Pd6zWjnsc7rKJD89xED33sYUDe17jDD9x7n70b79b78ljqTTWat/hjeV8fAK2SrEs=</latexit> <latexit sha1_base64="0UGu2/bvxwu5Z1zuLOLCgWfQP0c=">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</latexit> <latexit sha1_base64="0UGu2/bvxwu5Z1zuLOLCgWfQP0c=">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</latexit> <latexit sha1_base64="0UGu2/bvxwu5Z1zuLOLCgWfQP0c=">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</latexit> Convex objective: Symmetric matrices: Moment matrix: F <latexit sha1_base64="yNzr3sjH3uZAOlx/XwOw3O4OWh0=">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</latexit> <latexit sha1_base64="yNzr3sjH3uZAOlx/XwOw3O4OWh0=">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</latexit> <latexit sha1_base64="yNzr3sjH3uZAOlx/XwOw3O4OWh0=">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</latexit> <latexit sha1_base64="yNzr3sjH3uZAOlx/XwOw3O4OWh0=">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</latexit> H0, . . . , Hs 1 <latexit 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sha1_base64="zLcD9M76pgDcv01ZNB41ppKTIsU=">AAAC13icjVHLSsNAFD2Nr1pfsS7dBIvgopZEBF0W3HRZwT6kLSVJpzU0TUJmIpZS3Ilbf8Ct/pH4B/oX3hlTUIvohCRnzr3nzNx7ncj3uDDN14y2sLi0vJJdza2tb2xu6dv5Og+T2GU1N/TDuOnYnPlewGrCEz5rRjGzR47PGs7wTMYb1yzmXhhciHHEOiN7EHh9z7UFUV09X+maxbbfCwUvVroTfmhNu3rBLJlqGfPASkEB6aqG+gva6CGEiwQjMAQQhH3Y4PS0YMFERFwHE+JiQp6KM0yRI21CWYwybGKH9B3QrpWyAe2lJ1dql07x6Y1JaWCfNCHlxYTlaYaKJ8pZsr95T5SnvNuY/k7qNSJW4IrYv3SzzP/qZC0CfZyqGjyqKVKMrM5NXRLVFXlz40tVghwi4iTuUTwm7CrlrM+G0nBVu+ytreJvKlOycu+muQne5S1pwNbPcc6D+lHJMkvW+XGhXE5HncUu9nBA8zxBGRVUUSPvGzziCc/apXar3Wn3n6laJtXs4NvSHj4AJPCV/w==</latexit> Ms(m↵) : (m↵) 7! s 1 X ↵=0 m↵H↵ <latexit sha1_base64="TqvxfdxsVJ3rySWZ9KaD/fwGTFk=">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</latexit> <latexit 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sha1_base64="TqvxfdxsVJ3rySWZ9KaD/fwGTFk=">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</latexit> <latexit sha1_base64="TqvxfdxsVJ3rySWZ9KaD/fwGTFk=">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</latexit>
  39. CONVEX DUALITY Yohann DE CASTRO Convex Objective inf (m↵)2M s

    1 X ↵=0 f↵m↵ <latexit sha1_base64="gMjn+pzsyf317jyjeOcE7Dkj0g0=">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</latexit> <latexit 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sha1_base64="gMjn+pzsyf317jyjeOcE7Dkj0g0=">AAADCnicjVHLShxBFD3TMYma1yQu3TQOAbPI0B0CyUYYCIKbwAQcFWzTVJc1TmH1g65qQZr5g/xJdu7ErT+QlST5Af0Lb5U1oJGQVNPd5557z6m6dbNKSW2i6GcneDD38NHj+YXFJ0+fPX/RfflqS5dNzcWIl6qsdzKmhZKFGBlplNipasHyTInt7PCTzW8fiVrLstg0x5XYy9lBIceSM0NU2l1PZDFO29U8TZiqJuwNxUnOzIQzFX6eJrrJ0/YmtRZNv7b6bTwNx744nKnSbi/qR26F90HsQQ9+DcvuBRLsowRHgxwCBQxhBQZNzy5iRKiI20NLXE1IurzAFIukbahKUAUj9pC+BxTterag2Hpqp+a0i6K3JmWI16Qpqa4mbHcLXb5xzpb9m3frPO3Zjumfea+cWIMJsf/SzSr/V2d7MRjjo+tBUk+VY2x33Ls07lbsycNbXRlyqIizeJ/yNWHulLN7Dp1Gu97t3TKXv3SVlrUx97UNruwpacDxn+O8D7be9eOoH3953xsM/KjnsYwVrNI8P2CADQwxIu/v+IFf+B18C06C0+DspjToeM0S7qzg/BrZsqu0</latexit> <latexit sha1_base64="gMjn+pzsyf317jyjeOcE7Dkj0g0=">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</latexit> inf (m↵)2M F( s 1 X ↵=0 m↵H↵) <latexit sha1_base64="0UGu2/bvxwu5Z1zuLOLCgWfQP0c=">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</latexit> <latexit 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sha1_base64="0UGu2/bvxwu5Z1zuLOLCgWfQP0c=">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</latexit> <latexit 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sha1_base64="zLcD9M76pgDcv01ZNB41ppKTIsU=">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</latexit> <latexit sha1_base64="zLcD9M76pgDcv01ZNB41ppKTIsU=">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</latexit> Ms(m↵) : (m↵) 7! s 1 X ↵=0 m↵H↵ <latexit 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sha1_base64="TqvxfdxsVJ3rySWZ9KaD/fwGTFk=">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</latexit> <latexit sha1_base64="TqvxfdxsVJ3rySWZ9KaD/fwGTFk=">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</latexit> <latexit sha1_base64="TqvxfdxsVJ3rySWZ9KaD/fwGTFk=">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</latexit> inf (m↵) F(Ms(m↵)) : m = (m↵) 2 M, Am = b . <latexit 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sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">AAADHHicjVHLShxBFD12THzkNUmWbgoHYQQZukMgEhAmCJKNYCDjCJY01WXNTGH1g+7qgAzzK/mT7LILbg1ZuzEQ/8FbZYsmIkk13X3q3HtO1b03KYyubBj+nAkezD58NDe/sPj4ydNnz1svXu5WeV1K1Ze5ycu9RFTK6Ez1rbZG7RWlEmli1CA52nTxwWdVVjrPPtnjQh2kYpTpoZbCEhW3Blxnw3jSSWMuTDEWq1Oe6BGfbHW24+qGXeXsHWfpxg3DSMh4KuxYCsO21zh7n24kXjzla9241Q67oV/sLoga0EazdvLWD3AcIodEjRQKGSxhA4GKnn1ECFEQd4AJcSUh7eMKUyyStqYsRRmC2CP6jmi337AZ7Z1n5dWSTjH0lqRkWCFNTnklYXca8/HaOzv2Pu+J93R3O6Z/0nilxFqMif2X7jrzf3WuFosh1n0NmmoqPOOqk41L7bvibs5uVWXJoSDO4UOKl4SlV173mXlN5Wt3vRU+fu4zHev2ssmt8cvdkgYc/T3Ou2D3dTcKu9HHN+1erxn1PJawjA7N8y16+IAd9Mn7K87wGxfBl+Bb8D04uUoNZhrNK/yxgtNLAfywHA==</latexit> Primal
  40. CONVEX DUALITY Yohann DE CASTRO Fenchel Dual 8W , 9XW

    s.t. rF(XW ) = W , F?(W) = hXW , Wi F(XW ) <latexit sha1_base64="HCSjbi26o4hXozwXJT4otwKOgZU=">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</latexit> <latexit sha1_base64="HCSjbi26o4hXozwXJT4otwKOgZU=">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</latexit> <latexit sha1_base64="HCSjbi26o4hXozwXJT4otwKOgZU=">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</latexit> <latexit sha1_base64="HCSjbi26o4hXozwXJT4otwKOgZU=">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</latexit> F?(W) := sup X2S d hX, Wi F(X) <latexit sha1_base64="ooEoGszwBhM+qN6hh9a8NMT3mZ0=">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</latexit> <latexit sha1_base64="ooEoGszwBhM+qN6hh9a8NMT3mZ0=">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</latexit> <latexit sha1_base64="ooEoGszwBhM+qN6hh9a8NMT3mZ0=">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</latexit> <latexit sha1_base64="ooEoGszwBhM+qN6hh9a8NMT3mZ0=">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</latexit> 8X , 9WX s.t. rF?(WX) = X , F(X) = hX, WX i F?(W) <latexit sha1_base64="tEedp0KjwetFSeyS5KaZa6v9EM4=">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</latexit> <latexit sha1_base64="tEedp0KjwetFSeyS5KaZa6v9EM4=">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</latexit> <latexit sha1_base64="tEedp0KjwetFSeyS5KaZa6v9EM4=">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</latexit> <latexit sha1_base64="tEedp0KjwetFSeyS5KaZa6v9EM4=">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</latexit> Key Property 1
  41. 8X , 9WX s.t. rF?(WX) = X , F(X) =

    hX, WX i F?(WX ) <latexit sha1_base64="EM7EVx4q0YdaT2V/w7mV0OD/S6U=">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</latexit> <latexit sha1_base64="EM7EVx4q0YdaT2V/w7mV0OD/S6U=">AAADKnicjVHPa9RAGH2NP1rrr1WPXgYXYYU1JCLUi7AgFI8V3O5CU5dJdnYNnU3izEQsxf/I/8SbFylePejRq158M6ZQLaITkrzvfd97M998eaNL65LkeC06d/7CxfWNS5uXr1y9dr134+aurVtTqHFR69pMc2mVLis1dqXTatoYJVe5VpP84InPT14rY8u6eu4OG7W/ksuqXJSFdKRmPZUtaiO1FtNsOMxEpt5wTysms2kmbOxiUpXMtRTbLzLrpBkwc+9xKH7VyrnYHjDMtKyWWonp0OtMCO6fFsx6/SROwhJnQdqBPrq1U/c+IsMcNQq0WEGhgiPWkLB89pAiQUNuH0fkDFEZ8gpvsUltyyrFCkn2gN8lo72OrRh7TxvUBXfRfA2VAnepqVlniP1uIuTb4OzZv3kfBU9/tkP+885rRdbhJdl/6U4q/1fne3FY4FHooWRPTWB8d0Xn0oZb8ScXp7pydGjIeTxn3hAXQXlyzyJobOjd360M+S+h0rM+LrraFl/9KTng9M9xngW7D+I0idNnD/ujUTfqDdzGHQw4zy2M8BQ7GNP7Pb7hO35E76IP0XH06VdptNZpbuG3FX3+CXiItNA=</latexit> <latexit sha1_base64="EM7EVx4q0YdaT2V/w7mV0OD/S6U=">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</latexit> <latexit sha1_base64="EM7EVx4q0YdaT2V/w7mV0OD/S6U=">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</latexit> CONVEX DUALITY Yohann DE CASTRO Fenchel Dual 8W , 9XW s.t. rF(XW ) = W , F?(W) = hXW , Wi F(XW ) <latexit sha1_base64="HCSjbi26o4hXozwXJT4otwKOgZU=">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</latexit> <latexit sha1_base64="HCSjbi26o4hXozwXJT4otwKOgZU=">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</latexit> <latexit sha1_base64="HCSjbi26o4hXozwXJT4otwKOgZU=">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</latexit> <latexit sha1_base64="HCSjbi26o4hXozwXJT4otwKOgZU=">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</latexit> F?(W) := sup X2S d hX, Wi F(X) <latexit sha1_base64="ooEoGszwBhM+qN6hh9a8NMT3mZ0=">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</latexit> <latexit sha1_base64="ooEoGszwBhM+qN6hh9a8NMT3mZ0=">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</latexit> <latexit sha1_base64="ooEoGszwBhM+qN6hh9a8NMT3mZ0=">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</latexit> <latexit sha1_base64="ooEoGszwBhM+qN6hh9a8NMT3mZ0=">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</latexit> Key Property 1
  42. CONVEX DUALITY Yohann DE CASTRO Dual Cone Key Property 2

    M? := n c 2 Rs : 8m = (m↵)s 1 ↵=0 2 M , hm, ci 0 o <latexit sha1_base64="BlxT1HjkiGRlUOVvO2PbDBi7f5Q=">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</latexit> <latexit 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  43. CONVEX DUALITY Yohann DE CASTRO Dual Cone Key Property 2

    M? := n c 2 Rs : 8m = (m↵)s 1 ↵=0 2 M , hm, ci 0 o <latexit sha1_base64="BlxT1HjkiGRlUOVvO2PbDBi7f5Q=">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</latexit> <latexit 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  44. CONVEX DUALITY Yohann DE CASTRO Dual Cone Key Property 2

    M? := n c 2 Rs : 8m = (m↵)s 1 ↵=0 2 M , hm, ci 0 o <latexit sha1_base64="BlxT1HjkiGRlUOVvO2PbDBi7f5Q=">AAADRnicjVFNaxRBEK2Z+BFXo6uCHrw0LkKEdZkRQQkshHjxIkRxk0A6u/R0emeb9HzY3SOEYQ/+O/+B+AvUizfxanVlFtQg2sPMvH5V73VXVVYb7XySfIritQsXL11ev9K7em3j+o3+zVt7rmqsVBNZmcoeZMIpo0s18dobdVBbJYrMqP3s5HmI779T1umqfONPa3VUiLzUcy2FR2rWf88L4RdSGPZyyp0XdmvMd3TOW8l1STFbsNdTx9kWZz0+r6wwhhXjzWLGhakX4uGsPQPjZDlt3aN0uRKSKR8OOeNGlLlRrBhKbgnyXL1N6KDlrD9IRgktdh6kHRhAt3ar/kfgcAwVSGigAAUleMQGBDh8DiGFBGrkjqBFziLSFFewhB5qG8xSmCGQPcFvjrvDji1xHzwdqSWeYvC1qGTwADUV5lnE4TRG8YacA/s375Y8w91O8Z91XgWyHhbI/ku3yvxfXajFwxyeUQ0aa6qJCdXJzqWhroSbs1+q8uhQIxfwMcYtYknKVZ8ZaRzVHnorKP6FMgMb9rLLbeBruCUOOP1znOfB3uNRmozSV08G2zvdqNfhHtyHTZznU9iGF7ALE/T+HG1Ed6K78Yf4W/w9/nGWGked5jb8ttbgJ486vk8=</latexit> <latexit 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  45. CONVEX DUALITY Yohann DE CASTRO Lagrangian Analysis Primal inf (m↵)

    F(Ms(m↵)) : m = (m↵) 2 M, Am = b . <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit 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sha1_base64="3dT1X2CZ01CYV4O/q27328ZR2DI=">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</latexit> <latexit sha1_base64="3dT1X2CZ01CYV4O/q27328ZR2DI=">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</latexit> <latexit sha1_base64="3dT1X2CZ01CYV4O/q27328ZR2DI=">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</latexit> <latexit sha1_base64="3dT1X2CZ01CYV4O/q27328ZR2DI=">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</latexit> inf X,m sup W,c,z L = <latexit sha1_base64="KD7lidw83Kpzi7Mm/F4iMAkIJIo=">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</latexit> <latexit sha1_base64="KD7lidw83Kpzi7Mm/F4iMAkIJIo=">AAAC5nicjVHLSsNAFD3GV31XXbqJFsFFKYkIuhEKbly4ULC2YKVMplM7mBfJRKila3fuxK0/4FY/RfwD/QvvjBHUIjohyZlz7zkz914v9mWqHOdlxBodG5+YLExNz8zOzS8UF5dO0ihLuKjxyI+ShsdS4ctQ1JRUvmjEiWCB54u6d7Gn4/VLkaQyCo9VLxZnATsPZUdypohqFVebMuy0+o1yMGimWdzq18u8fDVoBkx1OfPtg91WseRUHLPsYeDmoIR8HUbFZzTRRgSODAEEQijCPhhSek7hwkFM3Bn6xCWEpIkLDDBN2oyyBGUwYi/oe06705wNaa89U6PmdIpPb0JKG+ukiSgvIaxPs008M86a/c27bzz13Xr093KvgFiFLrF/6T4z/6vTtSh0sGNqkFRTbBhdHc9dMtMVfXP7S1WKHGLiNG5TPCHMjfKzz7bRpKZ23Vtm4q8mU7N6z/PcDG/6ljRg9+c4h8HJZsV1Ku7RVqlazUddwArWsEHz3EYV+zhEjbyv8YBHPFld68a6te4+Uq2RXLOMb8u6fwe69ZzL</latexit> <latexit sha1_base64="KD7lidw83Kpzi7Mm/F4iMAkIJIo=">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</latexit> <latexit sha1_base64="KD7lidw83Kpzi7Mm/F4iMAkIJIo=">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</latexit> Primal (1) <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">AAACxnicjVHLSsNAFD2Nr1pfVZdugkWom5KIoMuCmy4r2gfUIsl0WkPzYjJRShH8Abf6aeIf6F94Z5yCWkQnJDlz7j1n5t7rp2GQScd5LVgLi0vLK8XV0tr6xuZWeXunnSW5YLzFkjARXd/LeBjEvCUDGfJuKrgX+SHv+OMzFe/ccpEFSXwpJynvR94oDoYB8yRRF1X38LpccWqOXvY8cA2owKxmUn7BFQZIwJAjAkcMSTiEh4yeHlw4SInrY0qcIBToOMc9SqTNKYtThkfsmL4j2vUMG9NeeWZazeiUkF5BShsHpEkoTxBWp9k6nmtnxf7mPdWe6m4T+vvGKyJW4obYv3SzzP/qVC0SQ5zqGgKqKdWMqo4Zl1x3Rd3c/lKVJIeUOIUHFBeEmVbO+mxrTaZrV731dPxNZypW7ZnJzfGubkkDdn+Ocx60j2quU3PPjyv1uhl1EXvYR5XmeYI6GmiiRd4jPOIJz1bDiq3cuvtMtQpGs4tvy3r4AM5Fj5k=</latexit> (2) <latexit sha1_base64="wCyI/bQo+qUHwFAFVt2hAvzzc+w=">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</latexit> <latexit sha1_base64="wCyI/bQo+qUHwFAFVt2hAvzzc+w=">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</latexit> <latexit sha1_base64="wCyI/bQo+qUHwFAFVt2hAvzzc+w=">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</latexit> <latexit 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  46. CONVEX DUALITY Yohann DE CASTRO Lagrangian Analysis Primal inf (m↵)

    F(Ms(m↵)) : m = (m↵) 2 M, Am = b . <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit 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sha1_base64="Jw0+lkp6PZfvkK+NGCHGYXoj8o0=">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</latexit> <latexit sha1_base64="Jw0+lkp6PZfvkK+NGCHGYXoj8o0=">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</latexit> <latexit sha1_base64="Jw0+lkp6PZfvkK+NGCHGYXoj8o0=">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</latexit> <latexit sha1_base64="Jw0+lkp6PZfvkK+NGCHGYXoj8o0=">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</latexit> <latexit sha1_base64="Jw0+lkp6PZfvkK+NGCHGYXoj8o0=">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</latexit> = F(X) hW, Xi hz, bi + s 1 X ↵=0 m↵ hW, H↵ i + (A > z)↵ c↵ <latexit sha1_base64="3dT1X2CZ01CYV4O/q27328ZR2DI=">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</latexit> <latexit sha1_base64="3dT1X2CZ01CYV4O/q27328ZR2DI=">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</latexit> <latexit sha1_base64="3dT1X2CZ01CYV4O/q27328ZR2DI=">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</latexit> <latexit sha1_base64="3dT1X2CZ01CYV4O/q27328ZR2DI=">AAADUHicjVFNbxMxEJ3d0FIW2qZw4MBlIUJK1CbaRZXgUikVEuqxSKSJ1G0jr+smVrwfsr1IaZRf0H/XC+o/aI/cuMHYdSBQIfBq7ec38549nrQUXOkouvb82oOV1Ydrj4LHT9Y3NutbT49UUUnKerQQhRykRDHBc9bTXAs2KCUjWSpYP528N/H+ZyYVL/JPelqyk4yMcn7OKdFIDeuXex+ag1Y7ESQfCRb2dwaJtPAndbGTOirYTlSVDWcJEeWY7EXz05lqx/NseEcEScpHzeCX1YELOPl2c/800UUZXrRcoE2Xpa1hvRF1IjvC+yB2oAFuHBb1L5DAGRRAoYIMGOSgEQsgoPA7hhgiKJE7gRlyEhG3cQZzCFBbYRbDDILsBOcR7o4dm+PeeCqrpniKwF+iMoTXqCkwTyI2p4U2Xllnw/7Ne2Y9zd2muKbOK0NWwxjZf+kWmf+rM7VoOId3tgaONZWWMdVR51LZVzE3D5eq0uhQImfwGcYlYmqVi3cOrUbZ2s3bEhu/sZmGNXvqciu4NbfEBsd/tvM+OHrTiaNO/HG30e26Vq/BC3gFTeznW+jCARxCD72/es+90HvpX/nf/O817y7Vdys8g99GLfgBgv/AQA==</latexit> inf X,m sup W,c,z L = <latexit sha1_base64="KD7lidw83Kpzi7Mm/F4iMAkIJIo=">AAAC5nicjVHLSsNAFD3GV31XXbqJFsFFKYkIuhEKbly4ULC2YKVMplM7mBfJRKila3fuxK0/4FY/RfwD/QvvjBHUIjohyZlz7zkz914v9mWqHOdlxBodG5+YLExNz8zOzS8UF5dO0ihLuKjxyI+ShsdS4ctQ1JRUvmjEiWCB54u6d7Gn4/VLkaQyCo9VLxZnATsPZUdypohqFVebMuy0+o1yMGimWdzq18u8fDVoBkx1OfPtg91WseRUHLPsYeDmoIR8HUbFZzTRRgSODAEEQijCPhhSek7hwkFM3Bn6xCWEpIkLDDBN2oyyBGUwYi/oe06705wNaa89U6PmdIpPb0JKG+ukiSgvIaxPs008M86a/c27bzz13Xr093KvgFiFLrF/6T4z/6vTtSh0sGNqkFRTbBhdHc9dMtMVfXP7S1WKHGLiNG5TPCHMjfKzz7bRpKZ23Vtm4q8mU7N6z/PcDG/6ljRg9+c4h8HJZsV1Ku7RVqlazUddwArWsEHz3EYV+zhEjbyv8YBHPFld68a6te4+Uq2RXLOMb8u6fwe69ZzL</latexit> <latexit sha1_base64="KD7lidw83Kpzi7Mm/F4iMAkIJIo=">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</latexit> <latexit sha1_base64="KD7lidw83Kpzi7Mm/F4iMAkIJIo=">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</latexit> <latexit sha1_base64="KD7lidw83Kpzi7Mm/F4iMAkIJIo=">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</latexit> Primal (1) <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> (2) <latexit sha1_base64="wCyI/bQo+qUHwFAFVt2hAvzzc+w=">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</latexit> <latexit sha1_base64="wCyI/bQo+qUHwFAFVt2hAvzzc+w=">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</latexit> <latexit sha1_base64="wCyI/bQo+qUHwFAFVt2hAvzzc+w=">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</latexit> <latexit sha1_base64="wCyI/bQo+qUHwFAFVt2hAvzzc+w=">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</latexit> (1) <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit> <latexit sha1_base64="QZo+vz1Dun0R+jiTO+iQuwzSwCk=">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</latexit>
  47. sup W,c,z inf X,m L = <latexit sha1_base64="let2dGKQIRCdDFQmNuuFw4XSRPU=">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</latexit> <latexit sha1_base64="let2dGKQIRCdDFQmNuuFw4XSRPU=">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</latexit>

    <latexit sha1_base64="let2dGKQIRCdDFQmNuuFw4XSRPU=">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</latexit> <latexit sha1_base64="let2dGKQIRCdDFQmNuuFw4XSRPU=">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</latexit> (2) <latexit sha1_base64="wCyI/bQo+qUHwFAFVt2hAvzzc+w=">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</latexit> <latexit 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  48. CONVEX DUALITY Yohann DE CASTRO Dual sup W 2S d,z2Rm

    hz, bi F ?(W) : (hW, H↵ i)↵ + A > z 2 M? . <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> inf (m↵) F(Ms(m↵)) : m = (m↵) 2 M, Am = b . <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> Primal When no duality gap: hz?, bi F?(W?) = F(Ms(m?)) <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> Complementarity leads to: hW ? , Ms(m ?)i hz ? , bi = s 1 X ↵=0 m ? ↵ (hW ? , H↵ i + (A > z ?)↵) = 0 <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> hW?, Ms(m?)i = hz?, bi = F(Ms(m?)) + F?(W?) and W? = rF(Ms(m?)) . <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">AAADW3icjVFbS8MwGP26eq23qfjkS3EIG47RiqAPCoIgvggKzglWR9rFrZheTFJBh3/DvyY++K7/wQe/ZBl4QTSl7cn5zjnJl4Q5i4X0vGerZI+Mjo1PTDpT0zOzc+X5hVORFTyizShjGT8LiaAsTmlTxpLRs5xTkoSMtsLrPVVv3VIu4iw9kXc5vUhIN42v4ohIpNrlx4CRtMuo27oMhCS8ftgW1WSAawHXtZ2h5t5owmFhv/pZXlvbH6CqCas5wU1BOkFCZI8nfZJ2HjThmPpOkJKQEfdbTFBvtMsVr+Hp4f4EvgEVMOMoKz9BAB3IIIICEqCQgkTMgIDA5xx88CBH7gL6yHFEsa5TeAAHvQWqKCoIstf47eLs3LApzlWm0O4IV2H4cnS6sIqeDHUcsVrN1fVCJyv2t+y+zlR7u8N/aLISZCX0kP3LN1T+16d6kXAFW7qHGHvKNaO6i0xKoU9F7dz91JXEhBw5hTtY54gj7Ryes6s9Qveuzpbo+qtWKlbNI6Mt4E3tEi/Y/36dP8HpesP3Gv7xRmV321z1BCzDClTxPjdhFw7gCJqY/W6tWGtWvfRi27ZjTw+kJct4FuHLsJc+AJmZxAo=</latexit> <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit> <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit> <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit>
  49. CONVEX DUALITY Yohann DE CASTRO Dual sup W 2S d,z2Rm

    hz, bi F ?(W) : (hW, H↵ i)↵ + A > z 2 M? . <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> inf (m↵) F(Ms(m↵)) : m = (m↵) 2 M, Am = b . <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">AAADHHicjVHLShxBFD12THzkNUmWbgoHYQQZukMgEhAmCJKNYCDjCJY01WXNTGH1g+7qgAzzK/mT7LILbg1ZuzEQ/8FbZYsmIkk13X3q3HtO1b03KYyubBj+nAkezD58NDe/sPj4ydNnz1svXu5WeV1K1Ze5ycu9RFTK6Ez1rbZG7RWlEmli1CA52nTxwWdVVjrPPtnjQh2kYpTpoZbCEhW3Blxnw3jSSWMuTDEWq1Oe6BGfbHW24+qGXeXsHWfpxg3DSMh4KuxYCsO21zh7n24kXjzla9241Q67oV/sLoga0EazdvLWD3AcIodEjRQKGSxhA4GKnn1ECFEQd4AJcSUh7eMKUyyStqYsRRmC2CP6jmi337AZ7Z1n5dWSTjH0lqRkWCFNTnklYXca8/HaOzv2Pu+J93R3O6Z/0nilxFqMif2X7jrzf3WuFosh1n0NmmoqPOOqk41L7bvibs5uVWXJoSDO4UOKl4SlV173mXlN5Wt3vRU+fu4zHev2ssmt8cvdkgYc/T3Ou2D3dTcKu9HHN+1erxn1PJawjA7N8y16+IAd9Mn7K87wGxfBl+Bb8D04uUoNZhrNK/yxgtNLAfywHA==</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> Primal When no duality gap: hz?, bi F?(W?) = F(Ms(m?)) <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> Complementarity leads to: hW ? , Ms(m ?)i hz ? , bi = s 1 X ↵=0 m ? ↵ (hW ? , H↵ i + (A > z ?)↵) = 0 <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">AAADYXicjVFdS+QwFL2dun6vVvfRl+IgdNh1aEXQl4GRffFlQWHHEaxT0hjHMukHSSq4w/yT/V/isy/6L/YmkwFXEU1pe3LuOSe5SVrxTKowfHAa7tyX+YXFpeWV1a9r697G5pksa0FZj5a8FOcpkYxnBeupTHF2XglG8pSzfjr6qev9WyZkVha/1V3FLnMyLLLrjBKFVOL9jTkphpz5/UEsFRE/fiUyyKe4FQtT251p/lhNagudWNZ5Mo4Jr25IJ5wMxnI3mlh3MqX94PUKx7ZiQ74HR4NYldUsvWXLrU6YeM2wHZrhvwWRBU2w46T07iGGKyiBQg05MChAIeZAQOJzARGEUCF3CWPkBKLM1BlMYBm9NaoYKgiyI/wOcXZh2QLnOlMaN8VVOL4CnT7soKdEnUCsV/NNvTbJmn0ve2wy9d7u8J/arBxZBTfIfuSbKT/r070ouIZD00OGPVWG0d1Rm1KbU9E79190pTChQk7jK6wLxNQ4Z+fsG480veuzJab+ZJSa1XNqtTU8613iBUevr/MtONtrR2E7Ot1vdrv2qhdhC7YhwPs8gC4cwwn0gDoNJ3AiZ6/x6C65nrs5lTYc6/kG/w136x9zD8bb</latexit> hW?, Ms(m?)i = hz?, bi = F(Ms(m?)) + F?(W?) and W? = rF(Ms(m?)) . <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit> <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit> <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit> <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit>
  50. CONVEX DUALITY Yohann DE CASTRO Dual sup W 2S d,z2Rm

    hz, bi F ?(W) : (hW, H↵ i)↵ + A > z 2 M? . <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> inf (m↵) F(Ms(m↵)) : m = (m↵) 2 M, Am = b . <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> Primal When no duality gap: hz?, bi F?(W?) = F(Ms(m?)) <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> Complementarity leads to: hW ? , Ms(m ?)i hz ? , bi = s 1 X ↵=0 m ? ↵ (hW ? , H↵ i + (A > z ?)↵) = 0 <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> <latexit sha1_base64="c7Wvwg6gOyuIwRab7k6ID+OquH0=">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</latexit> hW?, Ms(m?)i = hz?, bi = F(Ms(m?)) + F?(W?) and W? = rF(Ms(m?)) . <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit> <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit> <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit> <latexit sha1_base64="MW8eBvMqjuUf/kBogLZDrLIXQc8=">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</latexit> Key Property 1
  51. CONVEX DUALITY Yohann DE CASTRO Dual sup W 2S d,z2Rm

    hz, bi F ?(W) : (hW, H↵ i)↵ + A > z 2 M? . <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">AAADXnicjVFdaxNBFL2b1Vqjtam+CL4MhkKKcbsrBcWniCB9EepHuoVOs8xupsnQ2Q9mZoU25If4y8Q3H/VX6J3pBKpFdEKSM+eec+7cmbyRQps4/hp0whs3126t3+7eubtxb7O3df9Q160q+LioZa2Ocqa5FBUfG2EkP2oUZ2UueZqfvbb19BNXWtTVR3Pe8JOSzSpxKgpmkMp6n6lum2yRUlHRkpl5npMP2XRILq4Q7yflsktzMaMLKlk1k5xcDHOqHHz6ZkK1YWqQ7lDykpLBSpIO9zPKZDNnXrnjt09eTaipG9eCuB4Fk+TtZYxrs6TDqJv1+nEUu0Wug8SDPvh1UPe+AIUp1FBACyVwqMAglsBA4+cYEoihQe4EFsgpRMLVOSyhi94WVRwVDNkz/J3h7tizFe5tpnbuArtI/Cp0EthGT406hdh2I67eumTL/i174TLt2c7xP/dZJbIG5sj+y7dS/q/PzmLgFF64GQTO1DjGTlf4lNbdij05uTKVwYQGOYunWFeIC+dc3TNxHu1mt3fLXP27U1rW7guvbeGHPSU+cPLnc14Hh8+iJI6Sd3v90cg/9To8gscwwPd8DiPYhwMYY/bPYDuIgt3Ot3At3Ag3L6WdwHsewG8rfPgL5a7Ewg==</latexit> <latexit sha1_base64="c6k3soe3mUs8JN7TSm9JqksVioA=">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</latexit> inf (m↵) F(Ms(m↵)) : m = (m↵) 2 M, Am = b . <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> <latexit sha1_base64="T4LSG29D2KL4OekO3l1aWlmIjLM=">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</latexit> Primal When no duality gap: hz?, bi F?(W?) = F(Ms(m?)) <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit sha1_base64="jqvckFlknF/TagBzOozjkj5hO1w=">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</latexit> <latexit 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sha1_base64="XXIViMKzD7Nl3JEQomy2Wis4NqQ=">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</latexit> <latexit sha1_base64="XXIViMKzD7Nl3JEQomy2Wis4NqQ=">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</latexit> <latexit sha1_base64="XXIViMKzD7Nl3JEQomy2Wis4NqQ=">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</latexit> dµ(t) <latexit 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  52. OPTIMAL EXPERIMENTAL DESIGN Yohann DE CASTRO An Example: Optimal Designs

    inf (m↵) log det(Ms(m↵)) : m = (m↵) 2 M, m0 = 1 . <latexit sha1_base64="BWKk9kfvAVZuLFfb4fv2kbwwt5M=">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</latexit> <latexit 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sha1_base64="BWKk9kfvAVZuLFfb4fv2kbwwt5M=">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</latexit> <latexit sha1_base64="BWKk9kfvAVZuLFfb4fv2kbwwt5M=">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</latexit> F?(W) = s log det( W) and rF(X) = X 1 <latexit sha1_base64="4ptbwU5G+yd1BYJ9fDZfMJWr5UM=">AAADB3icjVHLTttAFD1xaQtpKWlZsrGIKoVFIruq1G4qRUVCLEEiDwmTaGxPgpWJbWbGCBTlA/iT7rqruu0PdAv8AfxF70wdiYcqOpbte88958zcuWEuEqU977riPFt6/uLl8kr11evVN2u1t++6KitkxDtRJjLZD5niIkl5Ryda8H4uOZuGgvfCybap9065VEmWHujznB9N2ThNRknENEHD2tedQaA0k43e1pemagYiGwcx141mbys4KVjsBpqf6RlL47nNg5SFgrk7jT7x+4NZ058Pa3Wv5dnlPg78MqijXHtZ7RIBYmSIUGAKjhSaYgEGRc8hfHjICTvCjDBJUWLrHHNUSVsQixODETqh75iywxJNKTeeyqoj2kXQK0np4j1pMuJJis1urq0X1tmg//KeWU9ztnP6h6XXlFCNY0Kf0i2Y/6szvWiM8Nn2kFBPuUVMd1HpUthbMSd373SlySEnzMQx1SXFkVUu7tm1GmV7N3fLbP3GMg1q8qjkFrg1p6QB+w/H+Tjofmj5Xsvf/1hvt8tRL2MDm2jQPD+hjV3soUPe3/Abl7hyLpzvzg/n51+qUyk167i3nF9/AKtUqI8=</latexit> <latexit sha1_base64="4ptbwU5G+yd1BYJ9fDZfMJWr5UM=">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</latexit> <latexit sha1_base64="4ptbwU5G+yd1BYJ9fDZfMJWr5UM=">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</latexit> <latexit sha1_base64="4ptbwU5G+yd1BYJ9fDZfMJWr5UM=">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</latexit> 
 
 W? = Ms(m?) 1 <latexit sha1_base64="J0szJDCpL4A64/uJWaJrnVAdIIs=">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</latexit> <latexit sha1_base64="J0szJDCpL4A64/uJWaJrnVAdIIs=">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</latexit> <latexit sha1_base64="J0szJDCpL4A64/uJWaJrnVAdIIs=">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</latexit> <latexit sha1_base64="J0szJDCpL4A64/uJWaJrnVAdIIs=">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</latexit> Z ⌦ (s p?(t))dµ?(t) = 0 <latexit sha1_base64="TXVc3KZC8IYJ+dMtf49vzhwqepA=">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</latexit> <latexit 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sha1_base64="TXVc3KZC8IYJ+dMtf49vzhwqepA=">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</latexit> <latexit sha1_base64="TXVc3KZC8IYJ+dMtf49vzhwqepA=">AAAC93icjVHLTttAFD2Y8ijlEeiyG6tRpbAgsiOkdoMUiQ27plJDkHAajZ1JMorHtmbGSFHEf7Bjh9jyA2zbP6j6B+1f9M7UVAWEyli2z5x7z5m598ZFKrQJgh8L3uKLpeWV1Zdrr9Y3Nrdq2zvHOi9VwrtJnubqJGaapyLjXSNMyk8KxZmMU96Lp4c23jvjSos8+2xmBe9LNs7ESCTMEDWotSKRmcE8+ij5mJ039F7xJdKGqYbZ3Y0kMxMl/WEky7/sQTCo1YNm4Jb/GIQVqKNanbz2HRGGyJGghARHBkM4BYOm5xQhAhTE9TEnThESLs5xjjXSlpTFKYMRO6XvmHanFZvR3npqp07olJReRUof70iTU54ibE/zXbx0zpZ9ynvuPO3dZvSPKy9JrMGE2P/p7jKfq7O1GIzwwdUgqKbCMba6pHIpXVfszf1/qjLkUBBn8ZDiinDilHd99p1Gu9ptb5mL/3SZlrX7pMot8cvekgYcPhznY3DcaoZBM/y0X2+3q1Gv4g3eokHzfI82jtBBl7wvcIuv+ObNvEvvyrv+k+otVJrXuLe8m98aO6NI</latexit> p ?(t) = s 1 X ↵=0 hMs(m ?) 1 , H↵ it ↵ <latexit 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  53. OPTIMAL EXPERIMENTAL DESIGN Yohann DE CASTRO An Example: Optimal Designs

    inf (m↵) log det(Ms(m↵)) : m = (m↵) 2 M, m0 = 1 . <latexit sha1_base64="BWKk9kfvAVZuLFfb4fv2kbwwt5M=">AAADJXicjVFNaxRBEH0ZzXeMqx5zaVwCG4jLjAiGQGDBi5dABDcJpMPY09u726Tng5meQFj29/hPvHmTBE9eFXKNZ6vbCVFDiD3MzOtX9V53VSWF0ZUNw68zwYOHs3PzC4tLyyuPVh+3njzdr/K6lKovc5OXh4molNGZ6lttjTosSiXSxKiD5OSNix+cqrLSefbenhXqOBWjTA+1FJaouPWB62wYTzppzIUpxmJjyhM94pMX3OQjPlC2sxtXN9ENzrY5S3duGEYGjKfCjqUwbHeTonG4E3mXKd/sxq122A39YrdB1IA2mrWXty7AMUAOiRopFDJYwgYCFT1HiBCiIO4YE+JKQtrHFaZYIm1NWYoyBLEn9B3R7qhhM9o7z8qrJZ1i6C1JybBOmpzySsLuNObjtXd27F3eE+/p7nZG/6TxSom1GBN7n+468391rhaLIbZ8DZpqKjzjqpONS+274m7O/qjKkkNBnMMDipeEpVde95l5TeVrd70VPv7dZzrW7WWTW+OHuyUNOPp3nLfB/stuFHajd6/avV4z6gWs4Tk6NM/X6OEt9tAn70+4xBV+Bh+Dz8GX4Px3ajDTaJ7hrxV8+wVANLOx</latexit> <latexit 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sha1_base64="BWKk9kfvAVZuLFfb4fv2kbwwt5M=">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</latexit> F?(W) = s log det( W) and rF(X) = X 1 <latexit 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 W? = Ms(m?) 1 <latexit sha1_base64="J0szJDCpL4A64/uJWaJrnVAdIIs=">AAAC33icjVHLSsNAFD3G97vqTjfBIuiiJRFBN0LBjRtBwVrB1jJJpzU0L2YmQimCO3fi1h9wq38j/oH+hXfGCGoRnZDk3HPvOTN3rpeGgVSO8zJkDY+Mjo1PTE5Nz8zOzRcWFk9kkgmfV/0kTMSpxyQPg5hXVaBCfpoKziIv5DWvu6fztUsuZJDEx6qX8kbEOnHQDnymiGoWlmvndamY2C0dNOV69BFsnPdL7lWzUHTKjln2IHBzUES+DpPCM+poIYGPDBE4YijCIRgkPWdw4SAlroE+cYJQYPIcV5gibUZVnCoYsV36dig6y9mYYu0pjdqnXUJ6BSltrJEmoTpBWO9mm3xmnDX7m3ffeOqz9ejv5V4RsQoXxP6l+6z8r073otDGjukhoJ5Sw+ju/NwlM7eiT25/6UqRQ0qcxi3KC8K+UX7es2000vSu75aZ/Kup1KyO/bw2w5s+JQ3Y/TnOQXCyWXadsnu0VaxU8lFPYAWrWKd5bqOCfRyiSt7XeMAjnixm3Vi31t1HqTWUa5bwbVn37/N5mWg=</latexit> <latexit 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  54. CONVEX DUALITY Yohann DE CASTRO • Generic proof • Rich

    duality interplay • Applies to LOTS of problems

  55. -5 0.8 -4 0.6 0.4 -3 x 2 0.2 -2

    0.5 0 x 1 -0.2 0 -1 -0.4 -0.5 0 Yohann DE CASTRO THANK YOU Thank You