Claire Boyer

(Sorbonne Université)

https://s3-seminar.github.io/seminars/claire-boyer/

Title — Sampling rates for l1 synthesis

Abstract — This work investigates the problem of signal recovery from undersampled noisy sub-Gaussian measurements under the assumption of a synthesis-based sparsity model. Solving the l1-synthesis basis pursuit allows to simultaneously estimate a coefficient representation as well as the sought-for signal. However, due to linear dependencies within redundant dictionary atoms it might be impossible to identify a specific representation vector, although the actual signal is still successfully recovered. We study both estimation problems from a non-uniform, signal-dependent perspective. By utilizing results from linear inverse problems and convex geometry, we identify the sampling rate describing the phase transition of both formulations, and propose a “tight” estimated upper-bound. This is a joint work with Maximilian März (TU Berlin), Jonas Kahn and Pierre Weiss (CNRS, Toulouse).

Biography — Claire Boyer est maîtresse de conférences à Sorbonne Université depuis 2016, et a été membre associé au Département de Mathématiques et Applications à l’ENS Ulm de 2017 à 2020. Ses domaines de recherche sont au carrefour du compressed sensing, des statistiques de grande dimension, de l’optimisation, des problèmes inverses et du machine learning. Durant les 4 dernières années, elle a principalement consacré ses travaux de recherche à la reconstruction d’objets à partir d’un faible nombre d’observations linéaires, faible devant la dimension de l’objet à reconstruire. L’objet d’intérêt pouvait aussi bien être un vecteur en dimension finie (on parle alors de compressed sensing), qu’une matrice avec entrées manquantes (on parle alors de complétion de matrice), qu’une mesure de Radon (on parle alors de “compressed sensing off-the- grid”). Plus récemment, elle s’est aussi intéressée aux problèmes d’apprentissage statistique lorsque les données peuvent être manquantes.

Transcript

1. Sampling rates for `1-synthesis Maximilian März, Claire Boyer, Jonas Kahn, Pierre Weiss
2. 2 / 30 Joint work with Maximilian März Jonas Kahn Pierre Weiss (TU Berlin) (IMT Toulouse) (IMT Toulouse)
3. 3 / 30 Outline 1. Introduction 2. A primer on convex geometry 3. Signal recovery Convex gauge for signal recovery Sampling rate for signal recovery 4. Upper Bounds on the Conic Gaussian Width
4. 4 / 30 Summary 1. Introduction 2. A primer on convex geometry 3. Signal recovery Convex gauge for signal recovery Sampling rate for signal recovery 4. Upper Bounds on the Conic Gaussian Width
5. 5 / 30 Setting Linear Noisy Measurements I Signal: x 0 2 Rn I Measurements: y 2 Rm of x 0 via the linear acquisition model y = Ax 0 + e, (1) where A 2 Rm⇥n is a Gaussian measurement matrix e 2 Rm models measurement noise with kek 2  ⌘ for some ⌘ 0
6. 5 / 30 Setting Linear Noisy Measurements I Signal: x 0 2 Rn I Measurements: y 2 Rm of x 0 via the linear acquisition model y = Ax 0 + e, (1) where A 2 Rm⇥n is a Gaussian measurement matrix e 2 Rm models measurement noise with kek 2  ⌘ for some ⌘ 0 Gaussian assumption I classical benchmark setup in CS I It allows us to determine the sampling rate of a convex program (i.e., the number of required measurements for successful recovery) by calculating the so-called Gaussian mean width
7. 6 / 30 The signal structure As for the signal x 0 I sparsity hardly satisfied in any real-world application I but sparse representations using specific transforms Gabor dictionaries, wavelet systems or data-adaptive representations Synthesis formulation There exists a matrix D 2 Rn⇥d and a low-complexity representation z 0 2 Rd such that x 0 can be “synthesized” as x 0 = D · z 0 . I D = [d 1 , . . . , dd ] is the dictionary I its columns are the dictionary atoms.
8. 6 / 30 The signal structure As for the signal x 0 I sparsity hardly satisfied in any real-world application I but sparse representations using specific transforms Gabor dictionaries, wavelet systems or data-adaptive representations Synthesis formulation There exists a matrix D 2 Rn⇥d and a low-complexity representation z 0 2 Rd such that x 0 can be “synthesized” as x 0 = D · z 0 . I D = [d 1 , . . . , dd ] is the dictionary I its columns are the dictionary atoms. I In this work, focus on the synthesis formulation instead of the analysis one.
9. 7 / 30 Visually
10. 7 / 30 Visually
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12. 7 / 30 Visually
13. 8 / 30 Synthesis basis pursuit for coefficient/signal recovery Synthesis basis pursuit for coefficient recovery ˆ Z := argmin z2Rd kzk 1 s.t. ky ADzk 2  ⌘. (BPcoef ⌘ ) D 2 Rn⇥d I when n = d, for instance D = Id (or any B.O.S) classical basis pursuit can recover any s-sparse vector z 0 w.h.p. if A is sub-Gaussian with m & s · log(2n/s)
14. 8 / 30 Synthesis basis pursuit for coefficient/signal recovery Synthesis basis pursuit for coefficient recovery ˆ Z := argmin z2Rd kzk 1 s.t. ky ADzk 2  ⌘. (BPcoef ⌘ ) D 2 Rn⇥d I when n = d, for instance D = Id (or any B.O.S) classical basis pursuit can recover any s-sparse vector z 0 w.h.p. if A is sub-Gaussian with m & s · log(2n/s) I in practice n ⌧ d, redundant D
15. 9 / 30 Synthesis basis pursuit for coefficient/signal recovery Synthesis basis pursuit for coefficient recovery ˆ Z := argmin z2Rd kzk 1 s.t. ky ADzk 2  ⌘. (BPcoef ⌘ ) D 2 Rn⇥d I when n = d, for instance D = Id (or any B.O.S) classical basis pursuit can recover any s-sparse vector z 0 w.h.p. if m & s · log(2n/s) I in practice n ⌧ d, redundant D representations not necessarily unique can’t expect to recover a specific representation via (BPcoef ⌘ ) One should be interested instead in: Synthesis basis pursuit for signal recovery ˆ X B D · argmin z2Rd kzk 1 s.t. ky ADzk 2  ⌘ ! | {z } =:ˆ Z . (BPsig ⌘ )
16. 10 / 30 Synthesis basis pursuit for coefficient/signal recovery In the noiseless case (i.e., when e = 0 and ⌘ = 0), I it might be the case that ˆ Z , {z 0 } (coefficient recovery fails) I but hope that ˆ X = D · ˆ Z = {x 0 } (signal recovery successes)
17. 10 / 30 Synthesis basis pursuit for coefficient/signal recovery In the noiseless case (i.e., when e = 0 and ⌘ = 0), I it might be the case that ˆ Z , {z 0 } (coefficient recovery fails) I but hope that ˆ X = D · ˆ Z = {x 0 } (signal recovery successes) Some questions addressed in the paper 1. When coefficient recovery , signal recovery? 2. How many measurements are required for coefficient recovery? signal recovery? 3. In case of coefficient and signal recovery, what about robustness to measurement noise? Questions addressed in this talk (Q1) What is the sampling rate for the signal recovery? (Q2) Which tight upper-bound can we provide on this sampling rate?
18. 11 / 30 Related works on the synthesis formulation [Rauhut, Schnass and Vandergheynst 2008] [Casazza, Chen, and Lynch, 2019] 7 Address the coefficient recovery and not the signal one
19. 11 / 30 Related works on the synthesis formulation [Rauhut, Schnass and Vandergheynst 2008] [Casazza, Chen, and Lynch, 2019] 7 Address the coefficient recovery and not the signal one Phase transitions of coefficient and signal recovery by `1-synthesis.
20. 11 / 30 Related works on the synthesis formulation [Rauhut, Schnass and Vandergheynst 2008] [Casazza, Chen, and Lynch, 2019] 7 Address the coefficient recovery and not the signal one 7 Uniform results over all s-sparse coefficient vectors
21. 11 / 30 Related works on the synthesis formulation [Rauhut, Schnass and Vandergheynst 2008] [Casazza, Chen, and Lynch, 2019] 7 Address the coefficient recovery and not the signal one 7 Uniform results over all s-sparse coefficient vectors
22. 11 / 30 Related works on the synthesis formulation [Rauhut, Schnass and Vandergheynst 2008] [Casazza, Chen, and Lynch, 2019] 7 Address the coefficient recovery and not the signal one 7 Uniform results over all s-sparse coefficient vectors 7 Rely on strong assumptions on D: RIP, NSP, incoherence ... 7 Forget about redundant representation systems highly coherent and with many linear dependencies
23. 11 / 30 Related works on the synthesis formulation [Rauhut, Schnass and Vandergheynst 2008] [Casazza, Chen, and Lynch, 2019] 7 Address the coefficient recovery and not the signal one 7 Uniform results over all s-sparse coefficient vectors 7 Rely on strong assumptions on D: RIP, NSP, incoherence ... 7 Forget about redundant representation systems highly coherent and with many linear dependencies Mission statement I Sampling rate for the signal recovery I Need for local and non-uniform approach: signal-dependent analysis is crucial for redundant representation systems I Avoiding strong assumptions on the dictionary
24. 12 / 30 Summary 1. Introduction 2. A primer on convex geometry 3. Signal recovery Convex gauge for signal recovery Sampling rate for signal recovery 4. Upper Bounds on the Conic Gaussian Width
25. 13 / 30 The generalized Basis Pursuit Consider the generalized basis pursuit min x2Rn f(x) s.t. ky Axk 2  ⌘, (BPf ⌘ ) f : Rn ! R is convex, supposed to reflect the “low complexity” of the signal x 0 .
26. 13 / 30 The generalized Basis Pursuit Consider the generalized basis pursuit min x2Rn f(x) s.t. ky Axk 2  ⌘, (BPf ⌘ ) f : Rn ! R is convex, supposed to reflect the “low complexity” of the signal x 0 . I The descent set of f at x 0 is given by D(f, x 0 ) := h 2 Rn : f(x 0 + h)  f(x 0 ) , (2) I The descent cone is defined by D^(f, x 0 ) := cone(D(f, x 0 )).
27. 13 / 30 The generalized Basis Pursuit Consider the generalized basis pursuit min x2Rn f(x) s.t. ky Axk 2  ⌘, (BPf ⌘ ) f : Rn ! R is convex, supposed to reflect the “low complexity” of the signal x 0 . I The descent set of f at x 0 is given by D(f, x 0 ) := h 2 Rn : f(x 0 + h)  f(x 0 ) , (2) I The descent cone is defined by D^(f, x 0 ) := cone(D(f, x 0 )). Definition (Minimum conic singular value) Consider A 2 Rm⇥n and a cone K ✓ Rn. The minimum conic singular value of A relative to K is: min (A, K) := inf x2K\Sn 1 kAxk2 . (3)
28. 14 / 30 A key quantity: Minimum conic singular value Consider the generalized basis pursuit min x2Rn f(x) s.t. ky Axk 2  ⌘, (BPf ⌘ ) f : Rn ! R is convex, supposed to reflect the “low complexity” of the signal x 0 . [Chandrasekaran et al. 2012, Tropp 2015] A deterministic error bound for (BPf ⌘ ) (a) If ⌘ = 0, exact recovery of x 0 by solving BPf ⌘=0 () min (A; D^(f, x 0 )) > 0 (b) In addition, any solution ˆ x of (BPf ⌘ ) satisfies x 0 ˆ x 2  2⌘ min (A; D^(f, x 0 )) . (4)
29. 14 / 30 A key quantity: Minimum conic singular value Consider the generalized basis pursuit min x2Rn f(x) s.t. ky Axk 2  ⌘, (BPf ⌘ ) f : Rn ! R is convex, supposed to reflect the “low complexity” of the signal x 0 . [Chandrasekaran et al. 2012, Tropp 2015] A deterministic error bound for (BPf ⌘ ) (a) If ⌘ = 0, exact recovery of x 0 by solving BPf ⌘=0 () min (A; D^(f, x 0 )) > 0 (b) In addition, any solution ˆ x of (BPf ⌘ ) satisfies x 0 ˆ x 2  2⌘ min (A; D^(f, x 0 )) . (4) f II Ha f Il.lk Xo no His a Jinx hotkey J'Az DQ.hn L'Keng DU.lk xo f II Ha f Il.lk Ho a no 4K a Jinx hotkey J'Aae Q.iypeo hotkey DU.lk xo O KA L t j'DH.t4xDierp H.lhieo
30. 14 / 30 A key quantity: Minimum conic singular value Consider the generalized basis pursuit min x2Rn f(x) s.t. ky Axk 2  ⌘, (BPf ⌘ ) f : Rn ! R is convex, supposed to reflect the “low complexity” of the signal x 0 . [Chandrasekaran et al. 2012, Tropp 2015] A deterministic error bound for (BPf ⌘ ) (a) If ⌘ = 0, exact recovery of x 0 by solving BPf ⌘=0 () min (A; D^(f, x 0 )) > 0 (b) In addition, any solution ˆ x of (BPf ⌘ ) satisfies x 0 ˆ x 2  2⌘ min (A; D^(f, x 0 )) . (4) I min (A; D^(f, x 0 )) can be NP-hard to compute I But there exists an estimate in the sub-Gaussian case! I Through the Gordon’s Escape Through a Mesh theorem
31. 15 / 30 From the minimum conic singular value to the conic mean width Definition (Mean width) I The mean width of a set K 2 Rn is w(K) = E sup h2K hg, hi ! with g ⇠ N(0, In). I The conic mean width of a cone K 2 Rn is w^(K) = w(K \ Sn 1) Theorem (Generic recovery) Assume that A 2 Rm⇥n is a Gaussian random matrix then min (A, K) p m 1 w^(K) u with probability larger than 1 e u2/2. A sufficient condition for robust recovery is m w^(D^(f, x 0 ))2 + 1.
32. 16 / 30 And actually, phase transitions [Amelunxen, Lotz, McCoy, Tropp (2014)] Theorem (Phase transitions) I For m w^(D^(f, x 0 ))2 + 1 log(✏) p n succeeds with probability > 1 ✏. I For m  w^(D^(f, x 0 ))2 + 1 + log(✏) p n succeeds with probability < ✏.
33. 17 / 30 Take-home messages on the generalized BP I Robust signal recovery via the generalized basis pursuit (BPf ⌘ ) is characterized by min (A; D^(f, x 0 )). I The required number of sub-Gaussian random measurements can be determined by the conic mean width of f at x 0 w2 ^ (D(f, x 0 )). I w2 ^ (D(f, x 0 )) gives a phase transition for the recovery success via BPf ⌘=0 , in the noiseless case. BPf ⌘=0 fails w.h.p. when m . w2 ^ (D(f, x 0 )) BPf ⌘=0 succeeds w.h.p. when m & w2 ^ (D(f, x 0 ))
34. 18 / 30 Summary 1. Introduction 2. A primer on convex geometry 3. Signal recovery Convex gauge for signal recovery Sampling rate for signal recovery 4. Upper Bounds on the Conic Gaussian Width
35. 19 / 30 Convex gauge for signal recovery Recall: synthesis basis pursuit for signal recovery ˆ X B D · argmin z2Rd kzk 1 s.t. ky ADzk 2  ⌘ ! . (BPsig ⌘ )
36. 19 / 30 Convex gauge for signal recovery Recall: synthesis basis pursuit for signal recovery ˆ X B D · argmin z2Rd kzk 1 s.t. ky ADzk 2  ⌘ ! . (BPsig ⌘ ) Let pK (x) := inf t>0 {x 2 tK} denote the gauge of a convex set K.
37. 19 / 30 Convex gauge for signal recovery Recall: synthesis basis pursuit for signal recovery ˆ X B D · argmin z2Rd kzk 1 s.t. ky ADzk 2  ⌘ ! . (BPsig ⌘ ) Let pK (x) := inf t>0 {x 2 tK} denote the gauge of a convex set K. Lemma (Gauge formulation) Assume that y = Ax 0 + e, with kek2  ⌘. Let D 2 Rn⇥d be a dictionary. Then, ˆ X = argmin x2Rn pD·Bd 1 (x) s.t. ky Axk 2  ⌘,
38. 19 / 30 Convex gauge for signal recovery Recall: synthesis basis pursuit for signal recovery ˆ X B D · argmin z2Rd kzk 1 s.t. ky ADzk 2  ⌘ ! . (BPsig ⌘ ) Let pK (x) := inf t>0 {x 2 tK} denote the gauge of a convex set K. Lemma (Gauge formulation) Assume that y = Ax 0 + e, with kek2  ⌘. Let D 2 Rn⇥d be a dictionary. Then, ˆ X = argmin x2Rn pD·Bd 1 (x) s.t. ky Axk 2  ⌘, Lemma (Descent cone) Let x 0 2 ran(D). For any z `1 2 Z `1 (`1-representers of x 0 in D), D^(pD·Bd 1 , x 0 ) = D · D^(k·k 1 , z `1 ) and D(pD·Bd 1 , x 0 ) = D · D(k·k 1 , z `1 ).
39. 20 / 30 Sampling rate for signal recovery Theorem (Signal recovery) Let D 2 Rn⇥d be a dictionary with x 0 2 ran(D) and pick any z `1 2 Z `1 . 8u > 0, with probability 1 e u2/2 : if m > m 0 B (w ^ (D · D(k·k 1 ; z `1 )) + u)2 + 1, (5) then any solution ˆ x to the program (BPsig ⌘=0 ) satisfies x 0 ˆ x 2  2⌘ p m 1 p m 0 1 . (6)
40. 20 / 30 Sampling rate for signal recovery Theorem (Signal recovery) Let D 2 Rn⇥d be a dictionary with x 0 2 ran(D) and pick any z `1 2 Z `1 . 8u > 0, with probability 1 e u2/2 : if m > m 0 B (w ^ (D · D(k·k 1 ; z `1 )) + u)2 + 1, (5) then any solution ˆ x to the program (BPsig ⌘=0 ) satisfies x 0 ˆ x 2  2⌘ p m 1 p m 0 1 . (6) (a) w2 ^ (D · D(k·k 1 ; z `1 )) drives the sampling rate (also true for coeff recovery)
41. 20 / 30 Sampling rate for signal recovery Theorem (Signal recovery) Let D 2 Rn⇥d be a dictionary with x 0 2 ran(D) and pick any z `1 2 Z `1 . 8u > 0, with probability 1 e u2/2 : if m > m 0 B (w ^ (D · D(k·k 1 ; z `1 )) + u)2 + 1, (5) then any solution ˆ x to the program (BPsig ⌘=0 ) satisfies x 0 ˆ x 2  2⌘ p m 1 p m 0 1 . (6) (a) w2 ^ (D · D(k·k 1 ; z `1 )) drives the sampling rate (also true for coeff recovery) (b) But the set of minimal `1-representers is not required to be a singleton: The descent cone in the signal space may be evaluated at any possible z `1 2 Z `1 .
42. 20 / 30 Sampling rate for signal recovery Theorem (Signal recovery) Let D 2 Rn⇥d be a dictionary with x 0 2 ran(D) and pick any z `1 2 Z `1 . 8u > 0, with probability 1 e u2/2 : if m > m 0 B (w ^ (D · D(k·k 1 ; z `1 )) + u)2 + 1, (5) then any solution ˆ x to the program (BPsig ⌘=0 ) satisfies x 0 ˆ x 2  2⌘ p m 1 p m 0 1 . (6) (a) w2 ^ (D · D(k·k 1 ; z `1 )) drives the sampling rate (also true for coeff recovery) (b) But the set of minimal `1-representers is not required to be a singleton: The descent cone in the signal space may be evaluated at any possible z `1 2 Z `1 . (c) Phase transition of signal recovery at m 0 .
43. 20 / 30 Sampling rate for signal recovery Theorem (Signal recovery) Let D 2 Rn⇥d be a dictionary with x 0 2 ran(D) and pick any z `1 2 Z `1 . 8u > 0, with probability 1 e u2/2 : if m > m 0 B (w ^ (D · D(k·k 1 ; z `1 )) + u)2 + 1, (5) then any solution ˆ x to the program (BPsig ⌘=0 ) satisfies x 0 ˆ x 2  2⌘ p m 1 p m 0 1 . (6) (a) w2 ^ (D · D(k·k 1 ; z `1 )) drives the sampling rate (also true for coeff recovery) (b) But the set of minimal `1-representers is not required to be a singleton: The descent cone in the signal space may be evaluated at any possible z `1 2 Z `1 . (c) Phase transition of signal recovery at m 0 . (d) Believe me, the robustness between signal and coefficient recovery is different
44. 21 / 30 Conclusion on these first results I Sampling rate for coeff rec = sampling rate for signal rec I (Robustness to noise is different) I Critical quantity = conic mean width of a linearly transformed cone w2 ^ (D · D(k·k 1 ; z `1 ))
45. 21 / 30 Conclusion on these first results I Sampling rate for coeff rec = sampling rate for signal rec I (Robustness to noise is different) I Critical quantity = conic mean width of a linearly transformed cone w2 ^ (D · D(k·k 1 ; z `1 )) Uniform recovery VS non uniform recovery Compressed sensing started with the Restricted Isometry Property leading to: All s-sparse vectors are recovered with probability X if m > mRIP . Here: A specific vector z 0 is recovered with probability Y if m > mz0 .
46. 21 / 30 Conclusion on these first results I Sampling rate for coeff rec = sampling rate for signal rec I (Robustness to noise is different) I Critical quantity = conic mean width of a linearly transformed cone w2 ^ (D · D(k·k 1 ; z `1 )) Uniform recovery VS non uniform recovery Compressed sensing started with the Restricted Isometry Property leading to: All s-sparse vectors are recovered with probability X if m > mRIP . Here: A specific vector z 0 is recovered with probability Y if m > mz0 . The Restricted Isometry Property... I RIP = far stronger statement. I RIP = optimal for orthogonal D, super pessimistic otherwise. I RIP = useless in 99% of the practical cases.
47. 22 / 30 Summary 1. Introduction 2. A primer on convex geometry 3. Signal recovery Convex gauge for signal recovery Sampling rate for signal recovery 4. Upper Bounds on the Conic Gaussian Width
48. 23 / 30 How to evaluate the conic mean width w2 ^ (D · D(k·k1 ; z`1 ))? 3 Tight and informative upper bounds for simple dictionaries such as orthogonal matrices 7 Involved for general, possibly redundant transforms 7 We cannot use classical argument based on polarity Indeed, 7 A bound based on a local condition number is too pessimistic w2 ^ (D · D(k·k 1 ; z `1 ) | {z } =:C )  kDk 2 min (D; D(k·k 1 ; z `1 )) · ⇣ w2 ^ (D(k·k 1 ; z `1 )) + 1 ⌘
49. 24 / 30 A geometric bound instead 1. Decompose the cone into its lineality and its range C = CL CR w2 ^ (C) . w2 ^ (CL ) + w2 ^ (CR ) + 1 2. The lineality CL is the largest subspace contained in the cone, so w2 ^ (CL ) ' dim(CL ) 3. The range is finitely generated, line-free, and contained into a circular cone of circumangle ↵ < ⇡/2 new bound on the conic mean width for such cones
50. 24 / 30 A geometric bound instead 1. Decompose the cone into its lineality and its range C = CL CR w2 ^ (C) . w2 ^ (CL ) + w2 ^ (CR ) + 1 2. The lineality CL is the largest subspace contained in the cone, so w2 ^ (CL ) ' dim(CL ) 3. The range is finitely generated, line-free, and contained into a circular cone of circumangle ↵ < ⇡/2 new bound on the conic mean width for such cones
51. 25 / 30 Decomposition of the descent cone of the gauge pD·Bd 1 Proposition Let D 2 Rn⇥d be a dictionary and let x 0 2 ran(D) \ {0}. Let C := D^(pD·Bd 1 , x 0 ) = D · D(k·k 1 ; z `1 ) denote the descent cone of the gauge at x 0 . Let z `1 2 ri(Z `1 ) be any minimal `1-representer of x 0 in D with maximal support and set ¯ S = supp(z `1 ) as well as ¯ s = # ¯ S. Assume ¯ s < d. Then we have: (a) The lineality space of C has a dimension less than ¯ s 1 and is given by CL = span(¯ s · sign(z `1,i ) · di D · sign(z `1 ) : i 2 ¯ S). (7) (b) The range of C is a 2(d ¯ s)-polyhedral ↵-cone given by: CR = cone(r±? j : j 2 ¯ Sc ) with r±? j B PC? L (±¯ s · dj D · sign(z `1 )) . (8)
52. 26 / 30 Circumangle for pointed polyhedral cone Proposition: Circumangle and circumcenter of polyhedral cones Let xi 2 Sn 1 for i 2 [k] and let C = cone(x 1 , . . . , xk ) be a nontrivial pointed polyhedral cone. Finding the circumcenter and circumangle of C amounts to solving the convex problem: cos(↵) = sup ✓2Bn 2 inf i2[k] h✓, xi i. 3 possible to numerically compute the circumangle of pointed polyhedral cones. , the minimum conic singular value is intractable in general
53. 27 / 30 Cmw for k-polyhedral cone contained into ↵-circular cones Proposition For k 5, the conic mean width of a k-polyhedral cone contained into an ↵-circular cone C in Rn is bounded by W(↵, k, n)  tan ↵ · 0 B B B B B B B B B B @ q 2 log ⇣ k/ p 2⇡ ⌘ + 1 q 2 log ⇣ k/ p 2⇡ ⌘ 1 C C C C C C C C C C A + 1 p 2⇡ . I the bound does not depend on the ambient dimension n, , in contrast to the conic width of a circular cone.
54. 28 / 30 Consequence for the sampling rate Theorem If ¯ s  d 3, we obtain that w2 ^ (D^(pD·Bd 1 , x 0 ))  ¯ s + 0 B B B B B B @tan ↵ · 0 B B B B B B @ s 2 log 2(d ¯ s) p 2⇡ ! + 1 1 C C C C C C A + 1 p 2⇡ 1 C C C C C C A 2 ,
55. 28 / 30 Consequence for the sampling rate Theorem If ¯ s  d 3, we obtain that w2 ^ (D^(pD·Bd 1 , x 0 ))  ¯ s + 0 B B B B B B @tan ↵ · 0 B B B B B B @ s 2 log 2(d ¯ s) p 2⇡ ! + 1 1 C C C C C C A + 1 p 2⇡ 1 C C C C C C A 2 , Corollary The critical number of measurements m 0 satisfies m 0 . ¯ s + tan2 ↵ · log(2(d ¯ s)/ p 2⇡). (9) The sampling rate is mainly governed by I the sparsity ¯ s of maximal support `1-representations of x 0 in D I the “narrowness” of the remaining cone CR , which is captured by its circumangle ↵ 2 [0, ⇡/2) I The number of dictionary atoms only has a logarithmic influence. NB: comparable to the mean width of a convex polytope, which is mainly determined by its diameter and by the logarithm of its number of vertices.
56. 29 / 30 Examples D x 0 2 Rn m & D = Id = 0 B B B B B B B B B B B B B @ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 C C C C C C C C C C C C C A s-sparse vector 2s log(2(n s)/ p 2⇡) 3 Convolutional dictionary 2-sparse (new) D = 0 B B B B B B B B B B B B @ 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 1 1 C C C C C C C C C C C C A (1 0 0 . . . 0 1)T 2 + 2 log(4n) : Total gradient variation Numerical evaluation D = r† s-gradient sparse s · log2(n) 3
57. 30 / 30 The end Contributions 3 Sampling rates for the synthesis problem (coefficient and signal). 3 Decent upper-bounds for the conic width of linearly transformed cones. 3 Dissected the descent cone of the `1-ball. 7 Quantities are still partly cryptic. 7 Case by case study of practical dictionaries is technical.
58. 30 / 30 The end Contributions 3 Sampling rates for the synthesis problem (coefficient and signal). 3 Decent upper-bounds for the conic width of linearly transformed cones. 3 Dissected the descent cone of the `1-ball. 7 Quantities are still partly cryptic. 7 Case by case study of practical dictionaries is technical. More to read in the paper arXiv:2004.07175
59. 30 / 30 The end Contributions 3 Sampling rates for the synthesis problem (coefficient and signal). 3 Decent upper-bounds for the conic width of linearly transformed cones. 3 Dissected the descent cone of the `1-ball. 7 Quantities are still partly cryptic. 7 Case by case study of practical dictionaries is technical. More to read in the paper arXiv:2004.07175 Thank you!