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
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
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
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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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
<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>
2 (t 1)2 8 (t2 1)2 <latexit sha1_base64="PL1HPibmFkFrEK63EhsulkoJqY4=">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</latexit> <latexit 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sha1_base64="PL1HPibmFkFrEK63EhsulkoJqY4=">AAAC5nicjVHLSgMxFD2O7/qqunQzWoS6aJmpgl0KblwqWBX6kJmYtkPnRSYjlNK1O3fi1h9wq58i/oH+hTdxCmoRzTDJybn3nOTmurHvJdKyXieMyanpmdm5+dzC4tLySn517SyJUsF4jUV+JC5cJ+G+F/Ka9KTPL2LBncD1+bnbO1Tx82suEi8KT2U/5s3A6YRe22OOJOoyv9loC4ftVkp6HRRlyd5pVYbVUlG2Khpf5gtW2dLDHAd2BgrIxnGUf0EDV4jAkCIARwhJ2IeDhL46bFiIiWtiQJwg5Ok4xxA50qaUxSnDIbZHc4d29YwNaa88E61mdIpPvyCliW3SRJQnCKvTTB1PtbNif/MeaE91tz6tbuYVECvRJfYv3SjzvzpVi0QbVV2DRzXFmlHVscwl1a+ibm5+qUqSQ0ycwlcUF4SZVo7e2dSaRNeu3tbR8TedqVi1Z1luind1S2qw/bOd4+CsUratsn2yVzioZq2ewwa2UKR+7uMARzhGjbxv8IgnPBtd49a4M+4/U42JTLOOb8N4+AChkppB</latexit> f(t) = <latexit 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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
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>
<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=">AAADh3icnVHJbtswEB1FabN0iZIcexFiFHAPdaUgSHIp4KCXXpoFqRfAcg2Kom0i1BKSKhAI/rn+RNA/aP8iQ0ZJF8OoUQqS3ryZ98jhxIXgSgfBd2fFXX3ydG19Y/PZ8xcvt7ztna7KS0lZh+Yil/2YKCZ4xjqaa8H6hWQkjQXrxVcfTL73lUnF8+yzvinYMCWTjI85JRqpkfctSomeUiL8T+8jwdOIZ+NR8ov8Ulks0+ry7HLWTN7MIlXGiml27UciybV6jJcSLS56u/8/3igaeY2gFdjlz4OwBg2o13nu3UIECeRAoYQUGGSgEQsgoPAZQAgBFMgNoUJOIuI2z2AGm6gtsYphBUH2Cr8TjAY1m2FsPJVVU9xF4CtR6cNr1ORYJxGb3XybL62zYRd5V9bTnO0G/3HtlSKrYYrsv3QPlcvqTC8axnBse+DYU2EZ0x2tXUp7K+bk/m9daXQokDM4wbxETK3y4Z59q1G2d3O3xOZ/2ErDmpjWtSX8NKfEAYd/j3MedPdbYdAKLw4a7XY96nV4BXvQxHkeQRs+wjl0gDpN59TpOX13w33nHrrH96UrTq3ZhT+We3IHDarX+g==</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 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sha1_base64="+/PjDPfIHft2PuPPDQdWvWG/hI8=">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</latexit> SDPs Non SDP
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
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]
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]