Matrix Completion and Recovery

CS 754 Lecture Notes

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Matrix Completion in Practice: Scenario 1

• Consider a survey of m people where each is asked q questions.
• It may not be possible to ask each person all q questions.
• Consider a matrix of size m by q (each row is the set of questions asked to any given person).
• This matrix is only partially filled (many missing entries).
• Is it possible to infer the full matrix given just the recorded entries?

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Matrix Completion in Practice: Scenario 2

• Some online shopping sites such as Amazon, Flipkart, Ebay, Netflix etc. have recommender systems.
• These websites collect product ratings from users (especially Netflix).
• Based on user ratings, these websites try to recommend other products/movies to the user that he/she will like with a high probability.
• Consider a matrix with the number of rows equal to the number of users, and number of columns equal to the number of movies/products.
• This matrix will be HIGHLY incomplete (no user has the patience to rate too many movies!!) – maybe only 5% of the entries will be filled up.
• Can the recommender system infer user preferences from just the defined entries?

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Matrix Completion in Practice: Scenario 2

• Read about the Netflix Prize to design a better recommender system:

http://en.wikipedia.org/wiki/Netflix_Prize

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Matrix Completion in Practice: Scenario 3

• Consider an image or a video with several pixel values missing.
• This is not uncommon in range imagery or remote sensing applications!
• Consider a matrix whose each column is a (vectorized) patch of m pixels. Let the number of columns be K.
• This m by K matrix will have many missing entries.
• Is it possible to infer the complete matrix given just the defined pixel values?
• If the answer were yes, note the implications for image compression!

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Matrix Completion in Practice: Scenario 4

• Consider a long video sequence of F frames.
• Suppose I mark out m salient (interesting points) {P_i}, 1<=i<=m, in the first frame.
• And track those points in all subsequent frames.
• Consider a matrix Μ of size m x 2F where row j contains the X and Y coordinates of the tracked points corresponding to initial point P_j (in each of the F frames).
• Unfortunately, many salient points may not be trackable due to occlusion or errors from the tracking algorithms.
• So Μ is highly incomplete.
• Is it possible to infer the true matrix from only the available measurements?

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�

• Given a set of images of a person’s face under L lighting conditions, create a matrix of size N x L where N is the number of pixels in an image.
• We are assuming that the person’s head pose does not change and that there are no shadows.
• It turns out that the rank of such a matrix is 3.
• For that we need to understand the so called Lambertian reflectance model in computer vision or computer graphics.

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Matrix Completion in Practice: Scenario 5

Lambertian Model

• The scene radiance is given by the equation:

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• What if cos(Ѳ) < 0? This means that the surface does not face the light source. In such a case, set the irradiance to 0.

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The term in the circle(s) is the effective surface area as seen from the light source

Many images of one person under different lighting directions: rank 3 matrix

• Consider

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If Lambertian model is satisfied!

M = number of pixels of the form (x,y)

m = number of images, each under a different lighting direction

Assumptions: no self-shadows or specularities

A property of these matrices

• Scenario 1: Many people will tend to give very similar or identical answers to many survey questions.
• Scenario 2: Many people will have similar preferences for movies (only a few factors affect user choices).
• Scenario 3: Non-local self-similarity!
• Scenario 4 and 5: Geometric constraints!
• This makes the matrices in all these scenarios (approximately) low in rank!

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Scenario 6: Low rank matrices!

• Consider matrix D whose entry D_ij = euclidean distance between points p_i and p_j in some k-dimensional space.
• Such a matrix has rank at the most k+2.
• Proof:

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https://infoscience.epfl.ch/record/196439/files/EPFL_TH5971.pdf

Column vector

A property of these matrices

• Scenario 4: The true matrix underlying Μ in question has been PROVED to be of low rank (in fact, rank 3) under orthographic projection and rigid motion (ref: Tomasi and Kanade, “Shape and Motion from Image Streams Under Orthography: a Factorization Method”, IJCV 1992) and a few other more complex camera models (up to rank 9).
• In case of orthographic projection with rigid motion, Μ can be expressed as a product of two matrices – a rotation matrix of size 2F x 3, and a shape matrix of size 3 x P. Hence it has rank 3.
• Μ is useful for many computer vision problems such as structure from motion, motion segmentation and multi-frame point correspondences.

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Low-rank matrices are cool!

• The answer to the four questions/scenarios is a NO in the general case.

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• But it’s a big YES if we assume that the underlying matrix has low rank (and which, as we have seen, is indeed the case for all four scenarios).

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• Low rank matrices of size n₁n₂ with rank r have only (n₁+n₂-r)r degrees of freedom – much less than n₁n₂ when r is small.

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Low-rank matrices are cool!

• Low rank matrices of size n₁n₂ with rank r have only (n₁+n₂-r)r degrees of freedom – much less than n₁n₂ when r is small.

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• #DOF for left singular vectors = n₁-1 for the first vector (as it is a unit vector) + n₁-2 for the second one (as it is a unit vector and orthogonal to the first one) + … + n₁-r (it’s a unit vector and orthogonal to the earlier r-1 singular vectors) = rn₁-r(r+1)/2
• #DOF for right singular vectors = rn₂-r(r+1)/2
• #DOF for singular values = r
• Total = rn₁+rn₂-r²

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Theorem 1 (Informal Statement)

• Consider an unknown matrix Μ of size n₁ by n₂ having rank r < min(n₁, n₂).
• Suppose we observe only a fraction of entries of Μ in the form of matrix Γ, where Γ(i,j) = Μ(i,j) for all (i,j) belonging to some set Ω and Γ(i,j) undefined elsewhere.
• If (1) Μ has row and column spaces that are “sufficiently incoherent” with the canonical basis (i.e. identity matrix), (2) r is “sufficiently small”, and (3) Ω is “sufficiently large”, then we can accurately recover Μ from Γ by solving the following rank minimization problem:

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Cool theorem, but … ☹

• The afore-mentioned optimization problem is NP-hard (in fact, current algorithms for solving it are known to have double exponential complexity in n₁ or n₂!)

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Theorem 2 (Informal Statement)

• Consider an unknown matrix Μ of size n₁ by n₂ having rank r < min(n₁, n₂).
• Suppose we observe only a fraction of entries of Μ in the form of matrix Γ, where Γ(i,j) = Μ(i,j) for all (i,j) belonging to some set Ω and Γ(i,j) undefined elsewhere.
• If (1) Μ has row and column spaces that are “sufficiently incoherent” with the canonical basis (i.e. identity matrix), (2) r is “sufficiently small”, and (3) Ω is “sufficiently large”, then we can accurately recover Μ from Γ by solving the following “trace-norm” minimization problem:

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E. Candes and B. Recht, “Exact matrix completion by convex optimization”, Found. Of Comp. Math., 2009

What is the trace-norm of a matrix?

• The trace-norm of a matrix is the sum of its singular values.
• It is also called nuclear norm.
• It is a softened version of the rank of a matrix, just like the L1-norm of a vector is a softened version of the L0-norm of the vector.
• Minimization of the trace-norm (even under the given constraints) is a convex optimization problem and can be solved efficiently (no local minima issues).
• This is similar to the L1-norm optimization (in compressive sensing) being efficiently solvable.

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More about trace-norm minimization

• The efficient trace-norm minimization procedure is provably known to give the EXACT SAME result as the NP-hard rank minimization problem (under the same constraints and same conditions on the unknown matrix Μ and the sampling set Ω).
• This is analogous to the case where L1-norm optimization yielded the same result as L0-norm optimization (under the same set of constraints and conditions).
• Henceforth we will concentrate only on Theorem 2 (and beyond).

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The devil is in the details

• Beware: Not all low-rank matrices can be recovered from partial measurements!
• Example consider a matrix containing zeroes everywhere except the top-right corner.
• This matrix is low rank, but it cannot be recovered from knowledge of only a fraction of its entries!
• Many other such examples exist.
• In reality, Theorems 1 and 2 work for low-rank matrices whose singular vectors are sufficiently spread out, i.e. sufficiently incoherent with the canonical basis (i.e. with the identity matrix).

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Coherence of a subspace

• The coherence of subspace U of Rⁿ of dimension r with respect to the canonical basis {e_i} is defined as:

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• Here P_U is the orthogonal projection onto U. It is given by P_U = U(U^TU)^-1U^T. This equals UU^T if U is an orthonormal basis. Note: U has size n x r.

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http://www.math.lsa.umich.edu/~speyer/417/OrthoProj.pdf

Coherence of a basis

• We are interested in matrices whose row and column subspaces (i.e. the left and right singular vectors respectively) have low coherence w.r.t. canonical basis.

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• Why? Because such matrices would not lie in the null-space of the sampling operator.

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• Remember: the sampling operator is a row-subsampled version of the identity matrix.

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Formal definition of key assumptions

• Consider an underlying matrix M of size n₁ by n₂. Let the SVD of M be given as follows:

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• We make the following assumptions about M:
• (A0)
• (A1) The max. abs. value entry in the n₁ by n₂ matrix is upper bounded by

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In absolute value

Theorem 2 (Formal Statement)

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the trace-norm minimizer (in the informal statement of theorem 2)

Candes and Recht , Exact matrix completion via convex optimization, https://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf

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Comments on Theorem 2

• Theorem 2 states that a larger number of entries of M (denoted by m) must be known for accurate reconstruction if (1) M has larger rank r, (2) greater value of μ₀ in (A0), (3) greater value of μ₁ in (A1).
• Example: If μ₀ = O(1) and the rank r is small, the reconstruction is accurate with high probability provided for constant C.

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Comments on Theorem 2

• It turns out that if the singular vectors of matrix M have bounded values, the condition (A1) almost always holds for the value μ₁ = O(log n).

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• The probabilistic nature of success of the recovery is because the entries in Ω are from a uniform distribution across all matrix (row,column) indices. Recovery is successful for most such subsets of Ω.

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Comments on Theorem 2

• Note that Ω cannot afford to completely miss any row or any column of M – otherwise recovery of even a rank-1 matrix is not possible.

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• Thus one needs at least one entry from every row and every column – and hence one needs at least n log n samples for this to happen.

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• The latter is an example of the coupon collectors problem – see https://en.wikipedia.org/wiki/Coupon_collector%27s_problem . This shows that the bound on m is not too conservative.

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Segway: Coupon Collector’s problem

• This problem is as follows: Consider n unique coupons, which you pick with replacement.

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• How many coupons do you need to draw with replacement to ensure that each coupon was drawn at least once?

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• The expected number of such trials is known to be O(n log n).

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• One can further show that the probability that the number of such trials exceeds cn log n (for some constant c) is upper bounded by n^-c+1.

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Another version of Theorem 2 = Theorem 2A

• Let M be a n₁ x n₂ matrix of rank r obeying assumption (A2) below, of which only m entries are observed with locations chosen uniformly at random. Consider m ≥ βC(μ_B)⁴n log²n where C is a constant and n = max(n₁,n₂). Then M is the unique solution to Q0 with probability 1-1/n^β. Here assumption (A2) is that:

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Candes and Plan, Matrix completion under noise, Proceedings of the IEEE

Matrix Completion under noise

• Consider an unknown matrix Μ of size n₁ by n₂ having rank r < min(n₁, n₂).
• Suppose we observe only a fraction of entries of Μ in the form of matrix Γ, where Γ(i,j) = Μ(i,j) + Z(i,j) for all (i,j) belonging to some set Ω and Γ(i,j) undefined elsewhere.
• Here Z refers to a white noise process which obeys the constraint that:

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Matrix Completion under noise

• In such cases, the unknown matrix Μ can be recovered by solving the following minimization procedure (called as a semi-definite program):

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Theorem 3

• Under the assumptions for Theorems 1,2,2A, the reconstruction result from (Q1) is accurate with an error bound given by:

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Candes and Plan, Matrix completion under noise, Proceedings of the IEEE

Numerical results: Exact matrix completion

• Test data (low rank matrices of rank r) generated synthetically using product of two random matrices of rank r each.
• Sample subset of entries uniformly at random.
• Estimation of underlying low-rank matrix using CVX.
• Study of variation of reconstruction error w.r.t. m and number of degrees of freedom (=r(2n-r) for a rank-r matrix).
• Recorded: probability of success in recovery across 50 trials.

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Candes and Recht , Exact matrix completion via convex optimization, https://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf

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Recall for n x n matrices, d_r = 2rn + r².

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Candes and Recht , Exact matrix completion via convex optimization, https://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf

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Candes and Recht , Exact matrix completion via convex optimization, https://statweb.stanford.edu/~candes/papers/MatrixCompletion.pdf

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This is actual a compressive low rank matrix recovery experiment, not a matrix completion experiment. See the section on (Compressive) Low Rank Matrix Recovery later on in these slides.

Numerical results: noisy matrix completion

• Test matrices created by product of two Gaussian random matrices of rank r.
• Sampling set Ω picked uniformly at random.
• Observations corrupted by Gaussian noise with mean 0 and σ = 1.
• Low rank matrix reconstructed as solution to:

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Numerical results: noisy matrix completion

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Red curve: proportional to upper bound on reconstruction error as per the theorem;

Oracle error in green curve: reconstruction error using least squares assuming that the matrix was of rank r

Numerical results: noisy matrix completion

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A Minimization Algorithm

• Consider the minimization problem:

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• There are many techniques to solve this problem (http://perception.csl.illinois.edu/matrix-rank/sample_code.html)
• Out of these, we will study one method called “singular value thresholding”.

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Singular Value Thresholding (SVT)

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Ref: Cai et al, A singular value thresholding algorithm for matrix completion, SIAM Journal on Optimization, 2010.

The soft-thresholding procedure obeys the following property (which we state w/o proof).

Properties of SVT (stated w/o proof)

• The sequence {Φ_k} converges to the true solution of the problem below provided the step-sizes {δ_k} all lie between 0 and 2.

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• For large values of τ, this converges to the solution of the original problem (i.e. without the Frobenius norm term).

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Properties of SVT (stated w/o proof)

• The matrices {Φ_k} turn out to have low rank (empirical observation – proof not established).
• The matrices {Y_k} also turn out to be sparse (empirical observation – rigorous proof not established).
• The SVT step does not require computation of full SVD – we need only those singular vectors whose singular values exceed τ. There are special iterative methods for that.

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Results

• The SVT algorithm works very efficiently and is easily implementable in MATLAB.

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• The authors report reconstruction of a 30,000 by 30,000 matrix in just 17 minutes on a 1.86 GHz dual-core desktop with 3 GB RAM and with MATLAB’s multithreading option enabled.

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Results (Data without noise)

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https://arxiv.org/abs/0810.3286

Results (Noisy Data)

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https://arxiv.org/abs/0810.3286

Results on real data

• Dataset consists of a matrix M of geodesic distances between 312 cities in the USA/Canada.
• This matrix is of approximately low-rank (in fact, the relative Frobenius error between M and its rank-3 approximation is 0.1159).
• 70% of the entries of this matrix (chosen uniformly at random) were blanked out.

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Results on real data

• The underlying matrix was estimated using SVT.
• In just a few seconds and a few iterations, the SVT produces an estimate that is as accurate as the best rank-3 approximation of M.

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Results on real data

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https://arxiv.org/abs/0810.3286

https://www.math.ust.hk/~jfcai/paper/SVT.pdf

Robust Principal Components Analysis

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Basic Question

• Consider a matrix M of size n1 x n2 that is the sum of two components – L (a low-rank component) and S (a component with sparse but unknown support).
• Can we recover L and S given only M?
• Note: support of S means all those matrix entries (i,j) where S(i,j) ≠ 0.

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Why ask this question? Scenario 1

• Consider a video taken with a static camera (like at airports or on overhead bridges on expressways).
• Such videos contain a “background” portion that is static or changes infrequently, and a moving portion called “foreground” (people, cars etc.).
• In many applications (esp. surveillance for security or traffic monitoring), we need to detect and track the moving foreground.

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Why ask this question? Scenario 1

• Let us represent this video as a N x T matrix, where the number of columns T equals the number of frames in the video, and where each column containing a single video-frame of N = N₁N₂ pixels converted to a vector form.
• This matrix can clearly be expressed as the sum of a low-rank background matrix and a sparse foreground matrix.

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Why ask this question? Scenario 2

• It is well-known that the images of a convex Lambertian object taken from the same viewpoint, but under K different lighting directions, span a low-dimensional space (ref: Basri and Jacobs, “Lambertian reflectance and linear subspaces”, IEEE TPAMI, 2003).
• This means that a matrix M of size N x K (each column of which is an image captured under one of the K lighting directions and represented in vectorized form) has a low approximate rank – in fact at most rank 9.

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Why ask this question? Scenario 2

• But many objects of interest (such as human faces) are neither convex nor exactly Lambertian.
• Effects such as shadows, specularities or saturation cause violation of the low-rank assumption.
• But these errors have sparse spatial support, even though their magnitudes are large.
• Here again, M can be represented as a low-rank background and a sparse foreground.

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Theorem 1 (Informal Statement)

• Consider a matrix M of size n₁ by n₂ which is the sum of a “sufficiently low-rank” component L and a “sufficiently sparse” component S whose support is uniformly randomly distributed in the entries of M.
• Then the solution of the following optimization problem (known as principal component pursuit) yields exact estimates of L and S with “very high” probability:

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This is a convex optimization problem.

Be careful!

• There is an implicit assumption in here that the low-rank component is not sparse (e.g.: a matrix containing zeros everywhere except the right top corner won’t do!).
• And that the sparse component is not low-rank (that’s why we said that it’s support is drawn from a uniform random distribution).

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Be careful!

• In addition, the low-rank component L must obey the assumptions A0 and A1 (required in matrix completion theory).

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Candes et al, Robust Principal Component Analysis?

https://statweb.stanford.edu/~candes/papers/RobustPCA.pdf

Theorem 1 (Formal Statement)

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Candes et al, Robust Principal Component Analysis?

https://statweb.stanford.edu/~candes/papers/RobustPCA.pdf

Comments on Theorem 1

• If the low-rank component L has “spread out” singular vectors so that μ is small, then larger and larger ranks for L can be accommodated.
• We require the support of S to be sparse and uniformly distributed but make no assumption about the magnitude/sign of the non-zero elements in S. But we don’t know the support of S.
• In the matrix completion problem, we knew which entries in the matrix were “missing”. Here we do not know the support of S. And the entries in S are not “missing” but unknown and possibly grossly corrupted!
• The probabilistic nature of the theorem is because the support of S is a uniformly randomly drawn subset of [1,2,…,n₁] x [1,2,…,n₂].
• Notice that there is no tuning parameter in the objective function.

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Matrix Recovery with Corrupted as well as missing data

• In some cases, we may have missing entries in the matrix (with known location) in addition to corrupted entries (whose location and magnitude we DON’T know).
• In such cases, we solve:

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Theorem 2 (Formal Statement)

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Candes et al, Robust Principal Component Analysis?

https://statweb.stanford.edu/~candes/papers/RobustPCA.pdf

Comments on Theorem 2

• If all entries were observed (i.e. m = n₁n₂), then this reduces to Theorem 1.
• If τ = 0, this becomes a pure matrix completion problem (no sparse component), but which is solved using a slightly different objective function (question: what’s that difference?).
• In the τ = 0 case, the minimization routine is guaranteed to give us a low-rank component as usual, but a “sparse” component that contains all zeros.
• In this theorem, observe that the guarantees hold only for L, and the recovery of S is not possible, as many entries of S may not be observed.
• The parameter for balancing between the L and S terms is auto-tuned and part of the mathematical analysis (no external tuning is required for the theorem to hold).

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Theorem 2.1: Noisy RPCA

• Under the same conditions on low-rank component L and support of sparse component S as in Theorem 2, and the same conditions on rank(L) and number of non-zero entries in S, the solution (L*,S*) of the following optimization problem Q2 satisfies the error bound

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where δ is such that

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Constant term

Zhou et al, Stable principal component pursuit, ISIT 2012

Numerical Results on RPCA

• Test matrices created by adding together low rank and sparse matrices.
• L matrices created from the product of low-rank Gaussian random matrices.
• S matrices created from uniform random sampling with entries marked 0,+1,-1 with probability 1-ρ, ρ/2, ρ/2.
• Success declared if relative error of low rank part is less than 0.001.
• A specific PCP algorithm is used.

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Numerical Results on RPCA

• In a second experiment, the sign of the sampled entries in S to be equal to the sign of the corresponding entries in L.
• In a third experiment, the S part is 0 and only L is estimated from its sampled version (i.e. the low rank matrix completion problem).

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Results

https://statweb.stanford.edu/~candes/papers/RobustPCA.pdf

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https://statweb.stanford.edu/~candes/papers/RobustPCA.pdf

Results on background subtraction

• Experiments on video sequences acquired in a shopping mall.
• Video size is 176 x 144 x 200, equal to a matrix of size 25344 x 200.
• Illumination change was minimal.
• Another experiment performed on a video of size 168 x 120 x 250 (matrix size 20,160 x 250), with severe illumination changes.

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Results on background subtraction

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https://statweb.stanford.edu/~candes/papers/RobustPCA.pdf

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https://statweb.stanford.edu/~candes/papers/RobustPCA.pdf

Results on shadow and specularity removal

• Experiments performed on images from the Yale face database.
• Image size is 192 x 168, images acquired under 58 different lighting conditions.
• Thus decomposition performed on a matrix of size 32,356 x 58.
• Each column of this matrix consists of a vectorized image of the same person from the same viewpoint but different lighting conditions.

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Algorithm for Robust PCA

• The algorithm uses the augmented Lagrangian technique.
• See Sec. 3.1 of https://web.stanford.edu/class/msande318/notes/notes09-BCL.pdf and https://www.him.uni-bonn.de/fileadmin/him/Section6_HIM_v1.pdf
• Suppose you want to solve:

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Algorithm for Robust PCA

• Suppose you want to solve:

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• The augmented Lagrangian method (ALM) adopts the following iterative updates:

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Augmentation term

Lagrangian term

ALM: Some intuition

• What is the intuition behind the update of the Lagrange parameters {λ_i}?

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• The problem is:

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The maximum w.r.t. λ will be ∞ unless the constraint is satisfied. Hence these problems are equivalent.

ALM: Some intuition

• The problem is:

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Due to non-smoothness of the max function, the equivalence has little computational benefit. We smooth it by adding another term that penalizes deviations from a prior estimate of the λ parameters.

Maximization w.r.t. λ is now easy

ALM: Some intuition – inequality constraints

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Maximization w.r.t. λ is now easy

Algorithm for Robust PCA

• In our case, we seek to optimize (µ > 0):

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• Basic algorithm:

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Lagrange matrix

Update of S using soft-thresholding

Update of L using singular-value soft-thresholding

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Alternating Minimization Algorithm for Robust PCA

Choice of µ is important. The algorithm in the paper simply sets it to 0.25 n₁ n₂/ ||M||_1. In practice, it should be set from a range of candidate values based on match between ||M-L”-S”||² and the noise level where L”, S” are estimates of L,S. This would require running the algorithm for many value of µ.

(Compressive) Low Rank Matrix Recovery

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Low rank matrix recovery

• Note: the third set of experiments in the section on matrix completion are not an example of matrix completion.

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• They are an example of a related problem: low rank matrix recovery, defined as:

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Low rank matrix recovery

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Low rank matrix recovery: restricted isometry property (RIP) for linear maps

• We consider A to be a linear map from R^n1xn2 to R^m.
• For integer r from 1 to min(n₁,n₂), the restricted isometry constant (RIC) of A of order r is the smallest number δ_r such that

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for all matrices M of rank at the most r.

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Low rank matrix recovery: Matrix restricted isometry property (RIP)

• What linear maps obey RIP?
• The following matrices obey RIP:
• Zero-mean iid Gaussian random matrices with variance 1/m
• (+1/-1)/√m Bernoulli random matrices

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Theorem 1 for low-rank matrix recovery

• If matrix A is such that δ_2r (A) < 1, then the equation A vec(M) = y has a unique solution where M is a matrix of rank at the most r.
• Proof by contradiction:

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Recht et al, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization”

Theorem 2 for low-rank matrix recovery

• If matrix A ∈ R^{m x n1n2} is such that δ_5r (A) < 0.1 for r ≥ 1 and M ∈ R^{n1 x n2} has rank at most r, then the solution to the following optimization problem is unique:

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Recht et al, “Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization”

Theorem 2 for low-rank matrix recovery

• This theorem has extensions to the case where M is approximately but not exactly of rank r.
• Approximately low rank means only a few singular values are large in value, and the others are zero or close to zero.
• And also to the case where the measurements in vector y are noisy.
• The proof of the result closely follows the proofs for compressive sensing recovery.

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Theorem 3 for low-rank matrix recovery

• Consider noisy measurements of the form:

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• Let M* be the solution to the following problem:

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• Then if δ_r(A) < 1/3, we have:

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The best rank-r approximation to M.

C₁ and C₂ are increasing functions of δ_r(A) in the domain [0,1/3].

Cai and Zhang, “Sharp RIP bound for sparse signal and low-rank matrix recovery”

Comments on Theorem 3

• It handles the case of noisy measurements as well as approximately (as opposed to exactly) low rank M.
• There is demonstration of a case where a matrix with δ_r(A) = 1/3 is unable to recover a set of low rank matrices M.

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Compressive RPCA: Algorithm and an Application

Primarily based on the paper:

Waters et al, “SpaRCS: Recovering Low-Rank and Sparse Matrices

from Compressive Measurements”, NIPS 2011

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Problem statement

• Let M be a matrix which is the sum of low rank matrix L and sparse matrix S.
• We observed compressive measurements of M in the following form:

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Scenarios

• M could be a matrix representing a video – each column of M is a vectorized frame from the video.
• M could also be a matrix representing a hyperspectral image – each column is the vectorized form of a slice at a given wavelength.
• Robust Matrix completion – a special form of a compressive L+S recovery problem.

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Objective function: SpaRCS

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Free parameters

SpaRCS = sparse and low rank decomposition via compressive sampling

SparCS Algorithm

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Very simple to implement; but requires tuning of K, r parameters; convergence guarantees not established.

https://papers.nips.cc/paper/4438-sparcs-recovering-low-rank-and-sparse-matrices-from-compressive-measurements.pdf

SparCS is similar to CoSamp

• CoSamp is a greedy algorithm for compressive inversion.
• The aforementioned SparCS algorithm is similar to CoSamp, but applied for compressive RPCA.
• For your reference, the CoSamp algorithm is given on the next slide.

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This is for compressive measurement vector u = 𝝓x

Results: Phase transition

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https://papers.nips.cc/paper/4438-sparcs-recovering-low-rank-and-sparse-matrices-from-compressive-measurements.pdf

Code: https://www.ece.rice.edu/~aew2/sparcs.html

p = #measurements

K = size of the support of S

r = rank of L

Results: Video CS

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Follows Rice SPC model, independent compressive measurements on each frame of the matrix M representing the video.

https://papers.nips.cc/paper/4438-sparcs-recovering-low-rank-and-sparse-matrices-from-compressive-measurements.pdf

Results: Video CS

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Follows Rice SPC model, independent compressive measurements on each frame of the matrix M representing the video.

https://papers.nips.cc/paper/4438-sparcs-recovering-low-rank-and-sparse-matrices-from-compressive-measurements.pdf

Results: Hyperspectral CS

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https://papers.nips.cc/paper/4438-sparcs-recovering-low-rank-and-sparse-matrices-from-compressive-measurements.pdf

Rice SPC model of CS measurements on every spectral band

Results: Robust matrix completion

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https://papers.nips.cc/paper/4438-sparcs-recovering-low-rank-and-sparse-matrices-from-compressive-measurements.pdf

Note that the support of S is a subset of Ω.

Theorem for Compressive PCP

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Wright et al, “Compressive Principal Component Pursuit”

http://yima.csl.illinois.edu/psfile/CPCP.pdf

Q is obtained from the linear span of different independent N(0,1) matrices with iid entries

This means that the #measurements >= #DOF of (L,S) times log²(m)

Theorem for Compressive PCP

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Wright et al, “Compressive Principal Component Pursuit”

http://yima.csl.illinois.edu/psfile/CPCP.pdf

Notion of μ-incoherent L0:

Definition of Q:

Comments

• The probabilistic nature of the theorem is due to randomness in the choice of Q, and also due to the sign of the values of S and the support of S.
• There is no assumption on the magnitude of the non-zero values in S.
• There is no randomness in L.
• #DOF(L) = mr + r + nr which is O(mr) as m > n. (Here m is not the number of measurements).
• #DOF(S) = ρmn.
• #required measurements for theoretical guarantees increases with ρ and r (besides m, n of course).
• Upper bound on r for the theoretical guarantees to hold is inversely proportional to µ.

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Summary

• Low rank matrix completion: motivation, key theorems, numerical results
• Algorithm for low rank matrix completion
• Robust PCA with algorithm and theoretical guarantees
• (Compressive) low rank matrix recovery with theoretical guarantees
• Compressive RPCA with theoretical guarantees and algorithm

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