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What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered An Overview of Compressed sensing M. Vidyasagar FRS Cecil & Ida Green Chair, The University of Texas at Dallas [email protected], www.utdallas.edu/∼m.vidyasagar Distinguished Professor, IIT Hyderabad [email protected] Ba;a;ga;va;tua;l .=+a;ma mUa;a;tRa and Ba;a;ga;va;tua;l Za;a:=+d;a;}ba Memorial Lecture University of Hyderabad, 16 March 2015 M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Compressed Sensing: Basic Problem Formulation Suppose x ∈ Rn is known to be k-sparse, where k n. That is, |supp(x)| ≤ k n, where supp denotes the support of a vector. However, it is not known which k components are nonzero. Can we recover x exactly by taking m n linear measurements of x? M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Precise Problem Statement: Basic Define the set of k-sparse vectors Σk = {x ∈ Rn : |supp(x)| ≤ k}. Do there exist an integer m, a “measurement matrix” A ∈ Rm×n , and a “demodulation map” ∆ : Rm → Rn , such that ∆(Ax) = x, ∀x ∈ Σk ? Note: Measurements are linear but demodulation map could be nonlinear. The algorithm is universal – the same A and ∆ need to work for every vector x, and nonadaptive – one has to choose all m rows of A at the outset. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Further Considerations What if: The vector x is not exactly sparse, but only nearly sparse? The measurement is corrupted by noise, and equals Ax + η The vector x is (nearly) sparse in some other basis, and not the canonical basis (e.g., time and frequency)? M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Some Definitions Given a norm k · k on Rn , for each integer k < n, define σk (x, k · k) := min kx − zk, z∈Σk the k-sparsity index of the vector x. x ∈ Σk (x is k-sparse) if and only if σk (x, k · k) = 0 for every norm. If x 6∈ Σk , then σk (x, k · k) depends on the specific norm. The k-sparsity index w.r.t. to an `p -norm is easy to compute. Let Λ0 denote the index set of the k largest components by magnitude of x. Then σk (x, k · kp ) = kxΛc0 kp . M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Precise Problem Statement: Final Given integers n, k n and a real number > 0, do there exist an integer m, a matrix A ∈ Rm×n , and a map ∆ : Rm → Rn such that k∆(Ax + η) − xk2 ≤ C1 σk (x, k · kp ) + C2 , whenever η ∈ Rm satisfies kηk2 ≤ ? Here C1 , C2 are “universal” constants that do not depend on x or η. If so the pair (A, ∆) is said to display near-ideal signal recovery. This formulation combines several desirable features into one. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Interpretation Suppose x is k-sparse, and suppose y = Ax + η. If an “oracle” knows the support set of x, call it J, then the standard least-squares estimate is x̂ = (AtJ AJ )−1 AtJ y = x + (AtJ AJ )−1 AtJ η, kx̂ − xk2 = k(AtJ AJ )−1 AtJ ηk2 ≤ C0 kηk2 , where C0 is the induced norm of (AtJ AJ )−1 AtJ . With a pair (A, ∆) that achieves near-ideal signal recovery, we have kx̂ − xk2 ≤ C2 kηk2 . So, if (A, ∆) achieves near-ideal signal recovery, then the estimation error is a constant multiple of that achievable by an “oracle,” but without knowing the support set of x. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Interpretation (Cont’d) If x is not sparse, but measurements are noise-free (η = 0), then the estimate x̂ = ∆(Ax) satisfies kx̂ − xk2 ≤ C1 σk (x, k · kp ). So the estimate is within a “universal constant” times the k-sparsity index of x. If x is k-sparse and measurements are noise-free, we get exact signal recovery. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Restricted Isometry Property Definition A matrix A ∈ Rm×n is said to satisfy the restricted isometry property (RIP) or order k with constant δk if (1 − δk )kuk22 ≤ kAuk22 ≤ (1 + δk )kuk22 , ∀u ∈ Σk . If J ⊆ {1, . . . , n} and |J| ≤ k, then the spectrum of AtJ AJ lies in the interval [1 − δk , 1 + δk ]. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Candeès-Tao Result on `1 -Norm Minimization Theorem Suppose √ A satisfies the RIP of order 2k with constant δ2k < 2 − 1. Given x ∈ Rn , define the demodulation map ∆ by ∆(y) = x̂ := arg min kzk1 s.t. Az = y. z Then ∆(Ax) = x, ∀x ∈ Σk . In plain English, any k-sparse vector can be recovered exactly by minimizing kzk1 for z in A−1 (Ax). Note that this is a linear programming problem. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered A More General Result Theorem m×n satisfies the RIP of order 2k with constant Suppose √A∈R δ2k < 2 − 1, and that y = Ax + η where kηk2 ≤ . Define x̂ = arg min kzk1 s.t. ky − Azk2 ≤ . z Then kx̂ − xk2 ≤ C0 σk (x, k · k1 ) √ + C2 , k where √ √ 1 + ( 2 − 1)δ2k 4 1 + δ2k √ √ C0 = 2 , C2 = . 1 − ( 2 + 1)δ2k 1 − ( 2 + 1)δ2k M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Some Observations Bounds are valid for all vectors x ∈ Rn – no assumptions of sparsity. The smaller we can make the RIP constant δ, the tighter the bounds. We suspect that making δ smaller requires larger values of m (more measurements). Indeed this is so, as we shall discover next. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Construction of Matrices with the RIP Challenge: Given integers n (dimension of√the vector), k (desired level of sparsity), and real number δ ∈ (0, 2 − 1), choose an integer m and a matrix A ∈ Rm×n such that A has the RIP of order 2k with constant δ. Refresher: A matrix A ∈ Rm×n is said to satisfy the restricted isometry property (RIP) or order k with constant δk if (1 − δk )kuk22 ≤ kAuk22 ≤ (1 + δk )kuk22 , ∀u ∈ Σk . Two distinct classes of approaches: deterministic and probabilistic. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Coherence of a Matrix Given a matrix A ∈ Rm×n , assume w.l.o.g. that it is column-normalized, that is each column has `2 -norm of one. Definition The one-column coherence µ1 (A) is defined as µ1 (A) := max max |hai , aj i|. i∈[n] j∈[n]\{i} The k-column coherence µk (A) is defined as X µk (A) := max max |hai , aj i|. i∈[n] S⊆[n]\{i},|S|≤k j∈S Coherence quantifies how “nearly orthonormal” the columns are. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Consequence of Low Coherence Easy consequence of the Gerschgorin circle theorem: Lemma Suppose A ∈ Rm×n is column-normalized. Let k ≤ m be a fixed integer. Suppose S ⊆ {1, . . . , n} and that |S| ≤ k ≤ m. Then spec(AtS AS ) ∈ [1 − µk−1 (A), 1 + µk−1 (A)]. Consequently A satisfies the RIP of order k with constant δk = µk−1 (A). M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study A Lower Bound on Coherence The following result is known as the “Welch bound.” Lemma Suppose A ∈ Rm×n is column-normalized; then r n−m 1 ≈√ , µ1 (A) ≥ m(n − 1) m r n−m k µk (A) ≥ k ≈√ , m(n − 1) m √ In view of the Welch bound, any matrix with µk (A) ≈ k/ m can be thought of as having “optimal coherence.” M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study A Typical Construction Due to DeVore (2007). Let p be a prime, and let a be a polynomial over the finite field Z/(p). Define the p × p matrix M (a) by 1 if j = a(i), [M (a)]ij = 0 if j 6= a(i). Note that each column of M (a) has one 1 and the rest 0. Define the p2 × 1 column vector ua by concatenating the p columns of M (a), and note that ua has exactly p ones and the rest are zero. 2 r+1 Now define A0 ∈ {0, 1}p ×p by lining up the columns M (a), as a varies over all pr+1 polynomials of degree r or less with √ coefficients in Z/(p). Finally, define A = A0 / p. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study A Typical Construction (Cont’d) Theorem The matrix A ∈ Rp 2 ×pr+1 has coherence µk (A) ≤ kr . p Note that p is the square root of the number of rows of A. So this construction is within a factor of r of an “optimally coherent” matrix. The matrix A is very sparse, with only 1/p elements being nonzero. √ The nonzero elements are all equal to 1/ p; so this matrix is “multiplication free.” M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Examples Given n, k, δ, we need to choose a prime number p such that (2k − 1)r (2k − 1)r 1/(r+1) r+1 ≤ δ, n ≤ p ⇐⇒ p ≥ max ,n . p δ We can choose r as we wish. Then m = p2 . Example 1: Let n = 10, 000, k = 5, δ = 0.4. Choosing r = 3 gives p ≥ 67.5 =⇒ p = 71 and m = p2 = 5, 041. Choosing r = 2 gives p ≥ 45 =⇒ p = 47 and m = p2 = 2, 209. Example 2: Let n = 106 , k = 10, δ = 0.4. Choosing r = 3 gives p ≥ 142.5 =⇒ p = 149 and m = p2 = 22, 201. Choosing r = 2 gives p ≥ 100 =⇒ p = 101 and m = p2 = 10, 201. In general we get m ≈ n2/3 unless k is very large. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Probabilistic Approach Let X 0 be a random variable with zero mean and standard √ deviation of one. Define X = X 0 / m, and define Φ ∈ Rm×n to consist of nm independent realizations of X. Then it is easy to see that E[kΦuk22 ] = kuk22 , ∀u ∈ Rn . If the r.v. kΦuk22 is also “highly concentrated” around its expected value, then “with high probability” the matrix Φ satisfies the RIP. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Sub-Gaussian Random Variables A r.v. X is said to be sub-Gaussian if there exist constants α, β such that Pr{|X| > t} ≤ α exp(−βt2 ), ∀t > 0. A normal r.v. satisfies the above with α = 2, β = 0.5. For later use, define c= β2 4α + 2β M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Main Result Set-up: Given a constant δ, choose any < δ (preferably very close to δ), and choose any r such that r 1+ r ≤1− . 1+δ Theorem Suppose X 0 is sub-Gaussian with constants α, β, and define c = β 2 /(4α + 2β). Define Φ as nm independent realizations of √ X 0 / m. Then Φ satisfies the RIP of order k with constant δ with probability at least equal to 1 − ζ, where ζ=2 en k 3 k k M. Vidyasagar FRS k exp(−mc2 ). Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Implementation of the Approach Suppose k, δ are given, and choose , r as above. To ensure that Φ satisfies the RIP of order k with constant δ with probability ≥ 1 − ζ, it suffices to choose 2 en 3 1 + k log m ≥ 2 log + k log c ζ k r samples of X. Note that m≈ k log n cδ 2 plus other terms. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Examples Choose X to be a Gaussian, so that α = 2, β = 0.5, and c = β 2 /(4α + 2β) = 1/44. Let ζ = 10−6 . Example 1: Let n = 10, 000, k = 5, δ = 0.4. Then m = 21, 683 > n. Compare with m = 2, 209 for the deterministic approach. Example 2: Let n = 106 , k = 10, δ = 0.4. Then m = 49, 863. Compare with m = 10, 201 for the deterministic approach. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Some General Considerations In the deterministic approach, m ≈ n2/(r+1) for some integer r, whereas in the probabilistic approach, m = O(k log n). But the O symbol hides a huge constant! In both cases, the probabilistic approach requires more measurements than the deterministic approach! Moreover, the deterministic approach leads to a highly sparse matrix, whereas the probabilistic approach leads to a matrix where every element is nonzero, with probability one. Open Problem: Find a deterministic approach that leads to m = O(k log n) measurements. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Time vs. Frequency Domain Whether a signal is “sparse” depends on the basis used. For example, a vector x denoting time samples of a signal may not be sparse, but its discrete cosine transform (or discrete Fourier transform) may be sparse. The use of the DFT requires measurement matrices with complex elements, but the theory works just the same. In particular, suppose M is the n × n discrete cosine transform matrix which is real and orthogonal, or the n × n discrete Fourier transform matrix which is complex and unitary. Transposes of “randomly selected” rows of these matrices satisfy the RIP. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Signal Reconstruction from Random Samples This example is due to Cleve Moler (2010). Suppose x(t) = sin(1394πt) + sin(3296πt), and we sample at 40KHz for 0.2 seconds (8,000 samples). The three frequencies involved (1394 Hz, 3296 Hz and 40,000 Hz) are not commensurate. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Discrete Cosine Transformation This signal is not sparse in the time domain! However, it is sparse in the frequency domain. For this purpose we employ the discrete cosine transform (dct). Given a vector x ∈ RN , its discrete cosine transform (dct) y ∈ RN is given by y(k) = w(k) N X x(n) cos n=1 π(2n − 1)(k − 1) 2N , k = [N ], where the weight vector w is defined by q 1 , k = 1, qN w(k) 2 N , k = 2, . . . , N. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Inverse Discrete Cosine Transformation Given a vector y ∈ RN , its inverse discrete cosine transform (idct) is given by x(n) = N X w(k)y(k) cos k=1 π(2n − 1)(k − 1) 2N , n = [N ], where the weight vector is as before. Both the dct and idct correspond to multiplying the vector by an orthogonal matrix. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Discrete Cosine Transform of Signal The dct of x(t) = sin(1394πt) + sin(3296πt) is shown below. It is highly concentrated around two frequencies, as expected. Spectra of Original and Approximated Signals 60 Original Approximated 40 Coefficient 20 0 −20 −40 −60 0 100 200 300 400 M. Vidyasagar FRS 500 600 Frequency 700 800 900 1000 Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Reconstruction of Signal from Random Samples There are 8,000 samples of the signal x(·). Now we will choose 500 samples at random from these 8,000 samples, and use those to reconstruct the signal. Note: When we generate 500 integers at random between 1 and 8,000, ony 485 distinct integers result. The next slide shows some of the samples. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Sampling of Signal at Random Locations The figure below shows the actual samples and some of the randomly chosen samples. Original Signal and Random Samples 2 1.5 Signal Value 1 0.5 0 −0.5 −1 −1.5 −2 0 0.002 0.004 0.006 0.008 Time in Seconds M. Vidyasagar FRS 0.01 0.012 0.014 Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Signal Reconstruction via `1 -Norm Minimization Let n = 8000, k = 485. Let S denote the index set of the randomly selected samples; note that |S| = 485 = k. Define D to be the dct of the n × n identity matrix. Thus the j-th column of D is the dct of the j-th elementary vector. Define A ∈ Rk×n to consist of the rows of D corresponding to the randomly selected samples; that is, A equals the projection of D onto the rows in S. Finally, let b ∈ Rk denote the randomly selected samples. We will reconstruct the original signal x ∈ Rn by setting x̂ = arg min kzk1 s.t. Az = b. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Reconstructed Signal in the Time Domain – 1 The figure below shows the original and reconstructed signal for small values of t. Reconstruction of a Signal with 8000 Samples Using 500 Frequencies 2 Original Reconstructed Original and Reconstructed Signals 1.5 1 0.5 0 −0.5 −1 −1.5 −2 0 0.005 0.01 0.015 0.02 0.025 Time M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Reconstructed Signal in the Time Domain – 2 The figure below shows the original and reconstructed signal for slightly larger values of t. The reconstruction is indistinguishable from the original signal. Reconstruction of a Signal with 8000 Samples Using 500 Frequencies 2 Original Reconstructed Original and Reconstructed Signals 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 0.025 0.03 0.035 0.04 0.045 0.05 Time M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Deterministic Approaches Probabilistic Approaches A Case Study Reconstructed Signal in the Frequency Domain The figure below shows the dcts of the original and reconstructed signals. Spectra of Original and Approximated Signals 60 Original Approximated 40 Coefficient 20 0 −20 −40 −60 0 100 200 300 400 M. Vidyasagar FRS 500 600 Frequency 700 800 900 1000 Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Outline 1 What is Compressed Sensing? 2 Main Results 3 Construction of Measurement Matrices Deterministic Approaches Probabilistic Approaches A Case Study 4 Some Topics Not Covered M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Group Sparsity Partition the index set {1, . . . , n} into g disjoint groups G1 , . . . , G g . A vector x ∈ Rn is “group k-sparse” if its support contains elements from very few groups, and |supp(x)| ≤ k. “Group” analogs of all previous results exist, e.g. group RIP (GRIP), and both exact recovery of group k-sparse vectors, as well as approximate recovery of nearly group k-sparse vectors. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Alternatives to the `1 -Norm What is so special about the `1 -norm? It turns out: Nothing! There are infinitely many norms that permit exact recovery of sparse or group-sparse vectors. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered One-Bit Compressed Sensing In standard compressed sensing, the measurement vector is y = Ax, or yi = hai , xi, for i = 1, . . . , m. What if yi = sign(hai , xi), for i = 1, . . . , m? This is called one-bit compressed sensing. The subject is still in its infancy. Perhaps the problem can be effectively analyzed using probably approximately correct (PAC) learning theory. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Low Rank Matrix Recovery Suppose M ∈ Rl×s has low rank, say ≤ k. By randomly sampling just m ls elements of M , is it possible to recover M exactly? Results to similar to vector recovery. Norm of a vector is replaced by “nuclear norm,” which is the sum of the singular values of a matrix. M. Vidyasagar FRS Overview of Compressed Sensing What is Compressed Sensing? Main Results Construction of Measurement Matrices Some Topics Not Covered Questions? M. Vidyasagar FRS Overview of Compressed Sensing