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Overview of the Mathematics of Compressive Sensing Simon Foucart Reading Seminar on “Compressive Sensing, Extensions, and Applications” Texas A&M University 29 October 2015 Optimality of Uniform Guarantees Gelfand Widths Gelfand Widths For a subset K of a normed space X , define linear m m m E (K , X ) := inf sup kx − ∆(Ax)k, A : X → R , ∆ : R → X x∈K Gelfand Widths For a subset K of a normed space X , define linear m m m E (K , X ) := inf sup kx − ∆(Ax)k, A : X → R , ∆ : R → X x∈K The Gelfand m-width of K in X is m m m d (K , X ) := inf sup kxk, L subspace of X , codim(L ) ≤ m x∈K ∩Lm Gelfand Widths For a subset K of a normed space X , define linear m m m E (K , X ) := inf sup kx − ∆(Ax)k, A : X → R , ∆ : R → X x∈K The Gelfand m-width of K in X is m m m d (K , X ) := inf sup kxk, L subspace of X , codim(L ) ≤ m x∈K ∩Lm If −K = K , then d m (K , X ) ≤ E m (K , X ), Gelfand Widths For a subset K of a normed space X , define linear m m m E (K , X ) := inf sup kx − ∆(Ax)k, A : X → R , ∆ : R → X x∈K The Gelfand m-width of K in X is m m m d (K , X ) := inf sup kxk, L subspace of X , codim(L ) ≤ m x∈K ∩Lm If −K = K , then d m (K , X ) ≤ E m (K , X ), and if in addition K + K ⊆ a K , then E m (K , X ) ≤ a d m (K , X ). Gelfand Widths of `1 -Balls: Upper Bound Gelfand Widths of `1 -Balls: Upper Bound Let A ∈ Rm×N with m ≈ c s ln(eN/m) be such that δ2s < 1/2. Gelfand Widths of `1 -Balls: Upper Bound Let A ∈ Rm×N with m ≈ c s ln(eN/m) be such that δ2s < 1/2. Let ∆1 : Rm → RN be the `1 -minimization map. Gelfand Widths of `1 -Balls: Upper Bound Let A ∈ Rm×N with m ≈ c s ln(eN/m) be such that δ2s < 1/2. Let ∆1 : Rm → RN be the `1 -minimization map. Given 1 < p ≤ 2, for any x ∈ B1N , kx − ∆1 (Ax)kp ≤ C s σ (x)1 ≤ 1−1/p s C s 1−1/p ≈ C0 . (m/ ln(eN/m))1−1/p Gelfand Widths of `1 -Balls: Upper Bound Let A ∈ Rm×N with m ≈ c s ln(eN/m) be such that δ2s < 1/2. Let ∆1 : Rm → RN be the `1 -minimization map. Given 1 < p ≤ 2, for any x ∈ B1N , kx − ∆1 (Ax)kp ≤ C s σ (x)1 ≤ 1−1/p s C s 1−1/p ≈ This gives an upper bound for E m (B1N , `N p ), C0 . (m/ ln(eN/m))1−1/p Gelfand Widths of `1 -Balls: Upper Bound Let A ∈ Rm×N with m ≈ c s ln(eN/m) be such that δ2s < 1/2. Let ∆1 : Rm → RN be the `1 -minimization map. Given 1 < p ≤ 2, for any x ∈ B1N , kx − ∆1 (Ax)kp ≤ C s σ (x)1 ≤ 1−1/p s C s 1−1/p ≈ C0 . (m/ ln(eN/m))1−1/p This gives an upper bound for E m (B1N , `N p ), and in turn d m (B1N , `N p) ln(eN/m) 1−1/p ≤ C min 1, . m Gelfand Widths of `1 -Balls: Lower Bound Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 for all x ∈ RN . s Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Then, ln(eN/m) 1−1/p C m N N c min 1, ≤ d m (B1N , `N p ) ≤ E (B1 , `p ) ≤ 1−1/p , m s Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Then, ln(eN/m) 1−1/p C m N N c min 1, ≤ d m (B1N , `N p ) ≤ E (B1 , `p ) ≤ 1−1/p , m s in other words ln(eN/m) 1 c 0 min 1, ≤ . m s Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Then, ln(eN/m) 1−1/p C m N N c min 1, ≤ d m (B1N , `N p ) ≤ E (B1 , `p ) ≤ 1−1/p , m s in other words ln(eN/m) 1 c 0 min 1, ≤ . m s 0 0 We derive either s ≤ 1/c or m ≥ c s ln(eN/m), Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Then, ln(eN/m) 1−1/p C m N N c min 1, ≤ d m (B1N , `N p ) ≤ E (B1 , `p ) ≤ 1−1/p , m s in other words ln(eN/m) 1 c 0 min 1, ≤ . m s 0 0 We derive either s ≤ 1/c or m ≥ c s ln(eN/m), in which case Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Then, ln(eN/m) 1−1/p C m N N c min 1, ≤ d m (B1N , `N p ) ≤ E (B1 , `p ) ≤ 1−1/p , m s in other words ln(eN/m) 1 c 0 min 1, ≤ . m s 0 0 We derive either s ≤ 1/c or m ≥ c s ln(eN/m), in which case eN s m ≥ c 0 s ln + c 0 s ln s m Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Then, ln(eN/m) 1−1/p C m N N c min 1, ≤ d m (B1N , `N p ) ≤ E (B1 , `p ) ≤ 1−1/p , m s in other words ln(eN/m) 1 c 0 min 1, ≤ . m s 0 0 We derive either s ≤ 1/c or m ≥ c s ln(eN/m), in which case eN s s m ≥ c 0 s ln + c 0 m ln s m m Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Then, ln(eN/m) 1−1/p C m N N c min 1, ≤ d m (B1N , `N p ) ≤ E (B1 , `p ) ≤ 1−1/p , m s in other words ln(eN/m) 1 c 0 min 1, ≤ . m s 0 0 We derive either s ≤ 1/c or m ≥ c s ln(eN/m), in which case eN s s m ≥ c 0 s ln + c0 m ln s |m {z m } ≥−1/e Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Then, ln(eN/m) 1−1/p C m N N c min 1, ≤ d m (B1N , `N p ) ≤ E (B1 , `p ) ≤ 1−1/p , m s in other words ln(eN/m) 1 c 0 min 1, ≤ . m s 0 0 We derive either s ≤ 1/c or m ≥ c s ln(eN/m), in which case eN s s m ≥ c 0 s ln + c0 m ln ≥ ··· s |m {z m } ≥−1/e Gelfand Widths of `1 -Balls: Lower Bound The Gelfand width of B1N in `N p , p > 1, also satisfies ln(eN/m) 1−1/p d m (B1N , `N ) ≥ c min 1, . p m Once established, this will show the optimality of the CS results. Indeed, suppose the existence of (A, ∆) such that C C kx − ∆(Ax)kp ≤ 1−1/p σs (x)1 ≤ 1−1/p kxk1 for all x ∈ RN . s s Then, ln(eN/m) 1−1/p C m N N c min 1, ≤ d m (B1N , `N p ) ≤ E (B1 , `p ) ≤ 1−1/p , m s in other words ln(eN/m) 1 c 0 min 1, ≤ . m s 0 0 We derive either s ≤ 1/c or m ≥ c s ln(eN/m), in which case eN s eN s m ≥ c 0 s ln + c0 m ln ≥ · · · ≥ c 00 s ln . s s |m {z m } ≥−1/e The lower estimate for d m (B1N , `Np ): two key insights The lower estimate for d m (B1N , `Np ): two key insights I small width implies `1 -recovery of s-sparse vectors for large s. The lower estimate for d m (B1N , `Np ): two key insights I small width implies `1 -recovery of s-sparse vectors for large s. I `1 -recovery of s-sparse vectors only possible for moderate s. The lower estimate for d m (B1N , `Np ): two key insights I small width implies `1 -recovery of s-sparse vectors for large s. There is a matrix A ∈ Rm×N such that every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax for s≈ I 1 m 2d (B1N , `N p) p p−1 . `1 -recovery of s-sparse vectors only possible for moderate s. The lower estimate for d m (B1N , `Np ): two key insights I small width implies `1 -recovery of s-sparse vectors for large s. There is a matrix A ∈ Rm×N such that every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax for s≈ I 1 m 2d (B1N , `N p) p p−1 . `1 -recovery of s-sparse vectors only possible for moderate s. For s ≥ 2, if A ∈ Rm×N is a matrix such that every s-sparse vector x is a minimizer of kzk1 subject to Az = Ax, then m ≥ c1 s ln N , c2 s The lower estimate for d m (B1N , `Np ): two key insights I small width implies `1 -recovery of s-sparse vectors for large s. There is a matrix A ∈ Rm×N such that every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax for s≈ I 1 m 2d (B1N , `N p) p p−1 . `1 -recovery of s-sparse vectors only possible for moderate s. For s ≥ 2, if A ∈ Rm×N is a matrix such that every s-sparse vector x is a minimizer of kzk1 subject to Az = Ax, then m ≥ c1 s ln N , c2 s c1 ≥ 0.45, c2 = 4. Deriving the lower estimate Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m ≤ 1. Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m Setting s ≈ 1/(cµ) ≥ 2, ≤ 1. Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m ≤ 1. Setting s ≈ 1/(cµ) ≥ 2, there exists A ∈ Rm×N allowing `1 -recovery of all s-sparse vectors. Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m ≤ 1. Setting s ≈ 1/(cµ) ≥ 2, there exists A ∈ Rm×N allowing `1 -recovery of all s-sparse vectors. Therefore m ≥ c1 s ln N c2 s Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m ≤ 1. Setting s ≈ 1/(cµ) ≥ 2, there exists A ∈ Rm×N allowing `1 -recovery of all s-sparse vectors. Therefore m ≥ c1 s ln N N ≥ c1 s ln 0 c2 s c2 m Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m ≤ 1. Setting s ≈ 1/(cµ) ≥ 2, there exists A ∈ Rm×N allowing `1 -recovery of all s-sparse vectors. Therefore m ≥ c1 s ln N N eN ≥ c1 s ln 0 ≥ c10 s ln c2 s c2 m m Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m ≤ 1. Setting s ≈ 1/(cµ) ≥ 2, there exists A ∈ Rm×N allowing `1 -recovery of all s-sparse vectors. Therefore N N eN ≥ c1 s ln 0 ≥ c10 s ln c2 s c2 m m 0 c ln(eN/m) ≥ 1 n ln(eN/m) o c min 1, m m ≥ c1 s ln Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m ≤ 1. Setting s ≈ 1/(cµ) ≥ 2, there exists A ∈ Rm×N allowing `1 -recovery of all s-sparse vectors. Therefore N N eN ≥ c1 s ln 0 ≥ c10 s ln c2 s c2 m m 0 0 c ln(eN/m) c ≥ 1 ≥ 1m n o ln(eN/m) c c min 1, m m ≥ c1 s ln Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m ≤ 1. Setting s ≈ 1/(cµ) ≥ 2, there exists A ∈ Rm×N allowing `1 -recovery of all s-sparse vectors. Therefore N N eN ≥ c1 s ln 0 ≥ c10 s ln c2 s c2 m m 0 0 c ln(eN/m) c ≥ 1 ≥ 1 m > m, n o ln(eN/m) c c min 1, m m ≥ c1 s ln provided c is chosen small enough. Deriving the lower estimate 1−1/p /2, where Suppose that d m (B1N , `N p ) < (cµ) ln(eN/m) µ := min 1, m ≤ 1. Setting s ≈ 1/(cµ) ≥ 2, there exists A ∈ Rm×N allowing `1 -recovery of all s-sparse vectors. Therefore N N eN ≥ c1 s ln 0 ≥ c10 s ln c2 s c2 m m 0 0 c ln(eN/m) c ≥ 1 ≥ 1 m > m, n o ln(eN/m) c c min 1, m m ≥ c1 s ln provided c is chosen small enough. This is a contradiction. Insight 1: width and `1 -recovery Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, i.e., kvS k1 ≤ kvS k1 for all v ∈ ker A and all S ∈ [N] with |S| ≤ s. Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, i.e., kvS k1 ≤ kvk1 /2 for all v ∈ ker A and all S ∈ [N] with |S| ≤ s. Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, i.e., kvS k1 ≤ kvk1 /2 for all v ∈ ker A and all S ∈ [N] with |S| ≤ s. m×N such that Setting d := d m (B1N , `N p ), there exists A ∈ R kvkp ≤ dkvk1 for all v ∈ ker A. Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, i.e., kvS k1 ≤ kvk1 /2 for all v ∈ ker A and all S ∈ [N] with |S| ≤ s. m×N such that Setting d := d m (B1N , `N p ), there exists A ∈ R kvkp ≤ dkvk1 for all v ∈ ker A. Then, for v ∈ ker A and S ∈ [N] with |S| ≤ s, Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, i.e., kvS k1 ≤ kvk1 /2 for all v ∈ ker A and all S ∈ [N] with |S| ≤ s. m×N such that Setting d := d m (B1N , `N p ), there exists A ∈ R kvkp ≤ dkvk1 for all v ∈ ker A. Then, for v ∈ ker A and S ∈ [N] with |S| ≤ s, kvS k1 ≤ s 1−1/p kvS kp Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, i.e., kvS k1 ≤ kvk1 /2 for all v ∈ ker A and all S ∈ [N] with |S| ≤ s. m×N such that Setting d := d m (B1N , `N p ), there exists A ∈ R kvkp ≤ dkvk1 for all v ∈ ker A. Then, for v ∈ ker A and S ∈ [N] with |S| ≤ s, kvS k1 ≤ s 1−1/p kvS kp ≤ s 1−1/p kvkp Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, i.e., kvS k1 ≤ kvk1 /2 for all v ∈ ker A and all S ∈ [N] with |S| ≤ s. m×N such that Setting d := d m (B1N , `N p ), there exists A ∈ R kvkp ≤ dkvk1 for all v ∈ ker A. Then, for v ∈ ker A and S ∈ [N] with |S| ≤ s, kvS k1 ≤ s 1−1/p kvS kp ≤ s 1−1/p kvkp ≤ ds 1−1/p kvk1 . Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, i.e., kvS k1 ≤ kvk1 /2 for all v ∈ ker A and all S ∈ [N] with |S| ≤ s. m×N such that Setting d := d m (B1N , `N p ), there exists A ∈ R kvkp ≤ dkvk1 for all v ∈ ker A. Then, for v ∈ ker A and S ∈ [N] with |S| ≤ s, kvS k1 ≤ s 1−1/p kvS kp ≤ s 1−1/p kvkp ≤ ds 1−1/p kvk1 . Choose s ≈ 1 p p−1 2d Insight 1: width and `1 -recovery Every s-sparse x ∈ RN is a minimizer of kzk1 subject to Az = Ax if and only if the null space property of order s holds, i.e., kvS k1 ≤ kvk1 /2 for all v ∈ ker A and all S ∈ [N] with |S| ≤ s. m×N such that Setting d := d m (B1N , `N p ), there exists A ∈ R kvkp ≤ dkvk1 for all v ∈ ker A. Then, for v ∈ ker A and S ∈ [N] with |S| ≤ s, kvS k1 ≤ s 1−1/p kvS kp ≤ s 1−1/p kvkp ≤ ds 1−1/p kvk1 . Choose s ≈ 1 p p−1 to derive the null space property of order s. 2d A combinatorial lemma A combinatorial lemma Lemma There exists n ≥ N s 2 4s subsets S 1 , . . . , S n of size s such that |S j ∩ S k | < s 2 for all 1 ≤ j 6= k ≤ n. A combinatorial lemma Lemma There exists n ≥ N s 2 4s subsets S 1 , . . . , S n of size s such that |S j ∩ S k | < s 2 for all 1 ≤ j 6= k ≤ n. Define s-sparse vectors x1 , . . . , xn by 1/s if i ∈ S j , j (x )i = 0 if i ∈ 6 Sj. A combinatorial lemma Lemma There exists n ≥ N s 2 4s subsets S 1 , . . . , S n of size s such that |S j ∩ S k | < s 2 for all 1 ≤ j 6= k ≤ n. Define s-sparse vectors x1 , . . . , xn by 1/s if i ∈ S j , j (x )i = 0 if i ∈ 6 Sj. Note that kxj k1 = 1 A combinatorial lemma Lemma There exists n ≥ N s 2 4s subsets S 1 , . . . , S n of size s such that |S j ∩ S k | < s 2 for all 1 ≤ j 6= k ≤ n. Define s-sparse vectors x1 , . . . , xn by 1/s if i ∈ S j , j (x )i = 0 if i ∈ 6 Sj. Note that kxj k1 = 1 and kxj − xk k1 > 1 for all 1 ≤ j 6= k ≤ n. Insight 2: `1 -recovery and number of measurements Insight 2: `1 -recovery and number of measurements For A ∈ Rm×N , suppose that every 2s-sparse vector x ∈ RN is a minimizer of kzk1 subject to Az = Ax. Insight 2: `1 -recovery and number of measurements For A ∈ Rm×N , suppose that every 2s-sparse vector x ∈ RN is a minimizer of kzk1 subject to Az = Ax. In the quotient space `N 1 / ker A, this means k[x]k := inf kx − vk1 v∈ker A Insight 2: `1 -recovery and number of measurements For A ∈ Rm×N , suppose that every 2s-sparse vector x ∈ RN is a minimizer of kzk1 subject to Az = Ax. In the quotient space `N 1 / ker A, this means k[x]k := inf kx − vk1 = kxk1 v∈ker A for all 2s-sparse x ∈ RN . Insight 2: `1 -recovery and number of measurements For A ∈ Rm×N , suppose that every 2s-sparse vector x ∈ RN is a minimizer of kzk1 subject to Az = Ax. In the quotient space `N 1 / ker A, this means k[x]k := inf kx − vk1 = kxk1 v∈ker A In particular, k[xj ]k = 1 for all 2s-sparse x ∈ RN . Insight 2: `1 -recovery and number of measurements For A ∈ Rm×N , suppose that every 2s-sparse vector x ∈ RN is a minimizer of kzk1 subject to Az = Ax. In the quotient space `N 1 / ker A, this means k[x]k := inf kx − vk1 = kxk1 v∈ker A for all 2s-sparse x ∈ RN . In particular, k[xj ]k = 1 and k[xj ] − [xk ]k > 1, all 1 ≤ j 6= k ≤ n. Insight 2: `1 -recovery and number of measurements For A ∈ Rm×N , suppose that every 2s-sparse vector x ∈ RN is a minimizer of kzk1 subject to Az = Ax. In the quotient space `N 1 / ker A, this means k[x]k := inf kx − vk1 = kxk1 v∈ker A for all 2s-sparse x ∈ RN . In particular, k[xj ]k = 1 and k[xj ] − [xk ]k > 1, all 1 ≤ j 6= k ≤ n. The size of this 1-separating set of the unit sphere satisfies 2 m n ≤ 1+ 1 Insight 2: `1 -recovery and number of measurements For A ∈ Rm×N , suppose that every 2s-sparse vector x ∈ RN is a minimizer of kzk1 subject to Az = Ax. In the quotient space `N 1 / ker A, this means k[x]k := inf kx − vk1 = kxk1 v∈ker A for all 2s-sparse x ∈ RN . In particular, k[xj ]k = 1 and k[xj ] − [xk ]k > 1, all 1 ≤ j 6= k ≤ n. The size of this 1-separating set of the unit sphere satisfies 2 m n ≤ 1+ = 3m . 1 Insight 2: `1 -recovery and number of measurements For A ∈ Rm×N , suppose that every 2s-sparse vector x ∈ RN is a minimizer of kzk1 subject to Az = Ax. In the quotient space `N 1 / ker A, this means k[x]k := inf kx − vk1 = kxk1 v∈ker A for all 2s-sparse x ∈ RN . In particular, k[xj ]k = 1 and k[xj ] − [xk ]k > 1, all 1 ≤ j 6= k ≤ n. The size of this 1-separating set of the unit sphere satisfies N s 2 4s 2 m ≤n ≤ 1+ = 3m . 1 Insight 2: `1 -recovery and number of measurements For A ∈ Rm×N , suppose that every 2s-sparse vector x ∈ RN is a minimizer of kzk1 subject to Az = Ax. In the quotient space `N 1 / ker A, this means k[x]k := inf kx − vk1 = kxk1 v∈ker A for all 2s-sparse x ∈ RN . In particular, k[xj ]k = 1 and k[xj ] − [xk ]k > 1, all 1 ≤ j 6= k ≤ n. The size of this 1-separating set of the unit sphere satisfies N s 2 4s 2 m ≤n ≤ 1+ = 3m . 1 Taking the logarithm yields m≥ N s ln . ln 9 4s