Please cite the publication as :
Donoho,
D.,
M.
Gavish,
and
A.
Montanari,
"The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising",
Stanford University.
Please cite the companion website as :
Donoho, D., M. Gavish, and A. Montanari, "The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising", RunMyCode companion website, http://www.execandshare.org/CompanionSite/Site265
Variable/Parameters  Description, constraint  Comments 

Case  'mat' for general matrices, 'sym' for symmetric positivesemidefinite matrices  
ρ  rank fraction, r/M for an MbyN matrix  
β  aspect ratio, M/N for MbyN matrix 
Variable/Parameters  Description  Visualisation 

Case  
ρ  
β 
Computing Date  Status  Actions 

Matan Gavish
Matan Gavish also created these companion sites
The Optimal Hard Threshold for Singular Values is 4/sqrt(3)
Abstract
We consider recovery of lowrank matrices from noisy data by hard thresholding of singular values, where singular values below a prescribed threshold \lambda are set to 0. We study the asymptotic MSE in a framework where the matrix size is large compared to the rank of the matrix to be recovered, and the signaltonoise ratio of the lowrank piece stays constant. The AMSEoptimal choice of hard threshold, in the case of nbyn matrix in noise level \sigma, is simply (4/\sqrt{3}) \sqrt{n}\sigma \approx 2.309 \sqrt{n}\sigma when \sigma is known, or simply 2.858\cdot y_{med} when \sigma is unknown, where y_{med} is the median empirical singular value. For nonsquare m by n matrices with m \neq n, these thresholding coefficients are replaced with different provided constants. In our asymptotic framework, this thresholding rule adapts to unknown rank and to unknown noise level in an optimal manner: it is always better than hard thresholding at any other value, no matter what the matrix is that we are trying to recover, and is always better than ideal Truncated SVD (TSVD), which truncates at the true rank of the lowrank matrix we are trying to recover. Hard thresholding at the recommended value to recover an nbyn matrix of rank r guarantees an AMSE at most 3nr\sigma^2. In comparison, the guarantee provided by TSVD is 5nr\sigma^2, the guarantee provided by optimally tuned singular value soft thresholding is 6nr\sigma^2, and the best guarantee achievable by any shrinkage of the data singular values is 2nr\sigma^2. Empirical evidence shows that these AMSE properties of the 4/\sqrt{3} thresholding rule remain valid even for relatively small n, and that performance improvement over TSVD and other shrinkage rules is substantial, turning it into the practical hard threshold of choice.
Gavish,
M.,
and
D.
Donoho,
"The Optimal Hard Threshold for Singular Values is 4/sqrt(3)",
Stanford University.
Authors:
Donoho
Gavish
Coders:
Gavish
Donoho Last update
05/30/2013
Ranking
9999
Runs
17
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Other Companion Sites on same paper
Other Companion Sites relative to similar papers
Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming
Abstract
Consider an underdetermined system of linear equations y = Ax with known d*n matrix
A and known y. We seek the sparsest nonnegative solution, i.e. the nonnegative x with fewest
nonzeros satisfying y = Ax. In general this problem is NPhard. However, for many matrices
A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be
found by linear programming. In classical convex polytope theory, a polytope P is called kneighborly if every set of k
vertices of P span a face of P. Let aj denote the jth column of A, 1<=_j<=_n, let a0 = 0 and
let P denote the convex hull of the aj . We say P is outwardly kneighborly if every subset
of k vertices not including 0 spans a face of P. We show that outward kneighborliness is
completely equivalent to the statement that, whenever y = Ax has a nonnegative solution
with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum.
Using this and classical results on polytope neighborliness we obtain two types of corollaries.
First, because many [d/2]neighborly polytopes are known, there are many systems
where the sparsest solution is available by convex optimization rather than combinatorial
optimization  provided the answer has fewer nonzeros than half the number of equations.
We mention examples involving incompletelyobserved Fourier transforms and Laplace transforms.
Second, results on classical neighborliness of highdimensional randomlyprojected simplices
imply that, if A is a typical uniformlydistributed random orthoprojector with n = 2d
and n large, the sparsest nonnegative solution to y = Ax can be found by linear programming
provided it has fewer nonzeros than 1/8 the number of equations.
We also consider a notion of weak neighborliness, in which the overwhelming majority
of ksets of aj's not containing 0 span a face. This implies that most nonnegative vectors
x with k nonzeros are uniquely determined by y = Ax. As a corollary of recent work
counting faces of random simplices, it is known that most polytopes P generated by large n
by 2n uniformlydistributed orthoprojectors A are weakly kneighborly with k _~.558n. We
infer that for most n by 2n underdetermined systems having a sparse solution with fewer
nonzeros than roughly half the number of equations, the sparsest solution can be found by
linear programming.
Donoho,
D.,
and
J.
Tanner,
"Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming ",
Proceedings of the National Academy of Sciences of the United States of America, 102, 9446–9451.
Authors:
Donoho
Tanner
Coders:
Donoho
Tanner Last update
10/08/2012
Ranking
17
Runs
6
Visits
81

Neighborliness of RandomlyProjected Simplices in High Dimensions
Abstract
Let A be a d by n matrix, d < n. Let T = T^n1 be the standard regular simplex in R^n. We count the faces of the projected simplex AT in the case where the projection is random, the dimension d is large and n and d are comparable: d ~ dn, d in (0, 1). The projector A is chosen uniformly at random from the Grassmann manifold of ddimensional orthoprojectors
of R^n. We derive ?N( d) > 0 with the property that, for any ? < ? N( deter), with overwhelming probability for large d, the number of kdimensional faces of P = AT is exactly the same as for T, for 0<=k<= ?d. This implies that P is [?d]neighborly, and its skeleton Skel[? d] ( P) is combinatorially equivalent to Skel[?d] (T). We display graphs of ?N. We also study a weaker notion of neighborliness it asks if the kfaces are all simplicial and if the numbers of kdimensional faces fk(P) >= fk(T)(1e). This was already considered by Vershik and Sporyshev, who obtained qualitative results about the existence of a threshold ? VS(d) > 0 at which phase transition occurs in k/d. We compute and display ?VS and compare to ?N.
Our results imply that the convex hull of n Gaussian samples in R^d, with n large and proportional to d, ‘looks like a simplex’ in the following sense. In a typical realization of such a highdimensional Gaussian point cloud d~ dn, all points are on the boundary of the convex hull, and all pairwise line segments, triangles, quadrangles, …, [?d]angles are on the boundary, for ?<? N(d/n).
Our results also quantify a precise phase transition in the ability of linear programming to find the sparsest nonnegative solution to typical systems of underdetermined linear equations; when there is a solution with fewer than ?VS(d/n)d nonzeros, linear programming will find that solution.
Donoho,
D.,
and
J.
Tanner,
"Neighborliness of RandomlyProjected Simplices in High Dimensions ",
Proceedings of the National Academy of Sciences of the United States of America , 102, 94529457.
Authors:
Donoho
Tanner
Coders:
Donoho
Tanner Last update
10/08/2012
Ranking
19
Runs
3
Visits
52

HighDimensional CentrallySymmetric Polytopes With Neighborliness Proportional to Dimension
Abstract
Let A be a d by n matrix, d < n. Let C be the regular cross polytope (octahedron) in
Rn. It has recently been shown that properties of the centrosymmetric polytope P = AC are
of interest for finding sparse solutions to the underdetermined system of equations y = Ax;
[9]. In particular, it is valuable to know that P is centrally kneighborly.
We study the face numbers of randomlyprojected crosspolytopes in the proportional dimensional
case where dn, where the projector A is chosen uniformly at random from
the Grassmann manifold of ddimensional orthoprojectors of Rn. We derive ?N(d) > 0 with
the property that, for any ? < ?N(d), with overwhelming probability for large d, the number
of kdimensional faces of P = AC is the same as for C, for 0 k d. This implies that
P is centrally bdcneighborly, and its skeleton Skel[? d](P) is combinatorially equivalent to
Skel[? d]©. We display graphs of ?N.
Two weaker notions of neighborliness are also important for understanding sparse solutions
of linear equations: facial neighborliness and sectional neighborliness [9]; we study
both. The weakest, (k, e)facial neighborliness, asks if the kfaces are all simplicial and if
the numbers of kdimensional faces fk(P) >= fk(C)(1  e). We characterize and compute
the critical proportion ?F (d) > 0 at which phase transition occurs in k/d. The other, (k, e)
sectional neighborliness, asks whether all, except for a small fraction epsilon, of the kdimensional
intrinsic sections of P are kdimensional crosspolytopes. (Intrinsic sections intersect P with
kdimensional subspaces spanned by vertices of P.) We characterize and compute a proportion
?S(d) > 0 guaranteeing this property for k/d ~ ? < ?S(d). We display graphs of ?S and
?F.
Donoho,
D.,
"HighDimensional CentrallySymmetric Polytopes With Neighborliness Proportional to Dimension",
Discrete & Computational Geometry, 35, 617652.
Authors:
Donoho
Coders:
Donoho
Last update
10/08/2012
Ranking
16
Runs
3
Visits
33

The Optimal Hard Threshold for Singular Values is 4/sqrt(3)
Abstract
We consider recovery of lowrank matrices from noisy data by hard thresholding of singular values, where singular values below a prescribed threshold \lambda are set to 0. We study the asymptotic MSE in a framework where the matrix size is large compared to the rank of the matrix to be recovered, and the signaltonoise ratio of the lowrank piece stays constant. The AMSEoptimal choice of hard threshold, in the case of nbyn matrix in noise level \sigma, is simply (4/\sqrt{3}) \sqrt{n}\sigma \approx 2.309 \sqrt{n}\sigma when \sigma is known, or simply 2.858\cdot y_{med} when \sigma is unknown, where y_{med} is the median empirical singular value. For nonsquare m by n matrices with m \neq n, these thresholding coefficients are replaced with different provided constants. In our asymptotic framework, this thresholding rule adapts to unknown rank and to unknown noise level in an optimal manner: it is always better than hard thresholding at any other value, no matter what the matrix is that we are trying to recover, and is always better than ideal Truncated SVD (TSVD), which truncates at the true rank of the lowrank matrix we are trying to recover. Hard thresholding at the recommended value to recover an nbyn matrix of rank r guarantees an AMSE at most 3nr\sigma^2. In comparison, the guarantee provided by TSVD is 5nr\sigma^2, the guarantee provided by optimally tuned singular value soft thresholding is 6nr\sigma^2, and the best guarantee achievable by any shrinkage of the data singular values is 2nr\sigma^2. Empirical evidence shows that these AMSE properties of the 4/\sqrt{3} thresholding rule remain valid even for relatively small n, and that performance improvement over TSVD and other shrinkage rules is substantial, turning it into the practical hard threshold of choice.
Gavish,
M.,
and
D.
Donoho,
"The Optimal Hard Threshold for Singular Values is 4/sqrt(3)",
Stanford University.
Authors:
Donoho
Gavish
Coders:
Gavish
Donoho Last update
05/30/2013
Ranking
9999
Runs
17
Visits
N.A.

Deterministic Matrices Matching the Compressed Sensing Phase Transitions of Gaussian Random Matrices
Abstract
In compressed sensing, one takes n < N samples of an N dimensional vector x0 using an n × N matrix A, obtaining undersampled measurements y = Ax0 . For random matrices with Gaussian i.i.d entries, it is known that, when x0 is ksparse, there is a precisely determined phase transition: for a certain region in the (k/n, n/N )phase diagram, convex optimization min x_1 subject to y = Ax, x ∈ X^N typically ﬁnds the sparsest solution, while outside that region, it typically fails. It has been shown empirically that the same property – with the same phase transition location – holds for a wide range of nonGaussian random matrix ensembles. We consider speciﬁc deterministic matrices including Spikes and Sines, Spikes and Noiselets, Paley Frames, DelsarteGoethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Extensive experiments show that for a typical ksparse object, convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian matrices. In our experiments, we considered coefﬁcients constrained to X^N for four different sets X ∈ {[0, 1], R_+ , R, C}. We establish this ﬁnding for each of the associated four phase transitions.
Monajemi,
H.,
D.
Donoho,
"Deterministic Matrices Matching the Compressed Sensing Phase Transitions of Gaussian Random Matrices",
Stanford University.
Authors:
Monajemi
Jafarpour Gavish Donoho
Coders:
Monajemi
Donoho Last update
01/04/2013
Ranking
9999
Runs
13
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86

Adaptive Estimation of Vector Autoregressive Models with TimeVarying Variance: Application to Testing Linear Causality in Mean
Abstract
Linear Vector AutoRegressive (VAR) models where the innovations could be unconditionally heteroscedastic and serially dependent are considered. The volatility structure is deterministic and quite general, including breaks or trending variances as special cases. In this framework we propose Ordinary Least Squares (OLS), Generalized Least Squares (GLS) and Adaptive Least Squares (ALS) procedures.
The GLS estimator requires the knowledge of the timevarying variance structure while in the ALS approach the unknown variance
is estimated by kernel smoothing with the outer product of the OLS residuals vectors. Different bandwidths for the different cells
of the timevarying variance matrix are also allowed. We derive the asymptotic distribution of the proposed estimators for the VAR
model coefficients and compare their properties. In particular we show that the ALS estimator is asymptotically equivalent to the
infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) in a given range and
hence justifies datadriven bandwidth rules. Using these results we build Wald tests for the linear Granger causality in mean which
are adapted to VAR processes driven by errors with a non stationary volatility. It is also shown that the commonly used standard Wald
test for the linear Granger causality in mean is potentially unreliable in our framework (incorrect level and lower asymptotic power).
Monte Carlo and realdata experiments illustrate the use of the different estimation approaches for the analysis of VAR models with timevarying variance innovations.
Raïssi,
H.,
"Adaptive Estimation of Vector Autoregressive Models with TimeVarying Variance: Application to Testing Linear Causality in Mean",
IRMARINSA and CREST ENSAI.
Authors:
Patilea
Coders:
Raïssi
Last update
10/08/2012
Ranking
58
Runs
9
Visits
169

Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise
Abstract
Overcomplete representations are attracting interest
in signal processing theory, particularly due to their potential to
generate sparse representations of signals. However, in general, the
problem of finding sparse representations must be unstable in the
presence of noise. This paper establishes the possibility of stable
recovery under a combination of sufficient sparsity and favorable
structure of the overcomplete system. Considering an ideal underlying
signal that has a sufficiently sparse representation, it is assumed
that only a noisy version of it can be observed. Assuming
further that the overcomplete system is incoherent, it is shown that
the optimally sparse approximation to the noisy data differs from
the optimally sparse decomposition of the ideal noiseless signal by
at most a constant multiple of the noise level. As this optimalsparsity
method requires heavy (combinatorial) computational effort,
approximation algorithms are considered. It is shown that similar
stability is also available using the basis and the matching pursuit
algorithms. Furthermore, it is shown that these methods result in
sparse approximation of the noisy data that contains only terms
also appearing in the unique sparsest representation of the ideal
noiseless sparse signal.
Donoho,
D.,
M.
Elad,
and
V.
Temlyakov,
"Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise",
Transactions on Information Theory, 52.
Authors:
Donoho
Elad Temlyakov
Coders:
Donoho
Elad Temlyakov Last update
10/08/2012
Ranking
14
Runs
10
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89

On the Stability of the Basis Pursuit in the Presence of Noise
Abstract
Given a signal S ( R^N and a fullrank matrix D ( R^NL with N<L, we define the signal’s overcomplete representation as a ( R^L satisfying S=Da. Among the infinitely many solutions of this underdetermined linear system of equations, we have special interest in the sparsest representation, i.e., the one minimizing a0. This problem has a combinatorial flavor to it, and its direct solution is impossible even for moderate L. Approximation algorithms are thus required, and one such appealing technique is the basis pursuit (BP) algorithm. This algorithm has been the focus of recent theoretical research effort. It was found that if indeed the representation is sparse enough, BP finds it accurately. When an error is permitted in the composition of the signal, we no longer require exact equality S=Da. The BP has been extended to treat this case, leading to a denoizing algorithm. The natural question to pose is how the abovementioned
theoretical results generalize to this more practical mode of operation. In this paper we propose such a generalization. The behavior of the basis pursuit in the presence of noise has been the subject of two independent very wide contributions released for publication very recently. This paper is another contribution in this direction, but as opposed to the others mentioned, this paper aims to present a somewhat simplified picture of the topic, and thus could be referred to as a primer to this field. Specifically, we establish here the stability of the BP in the presence of noise for sparse enough representations. We study both the case of a general dictionary D, and a special case where D is built as a union of orthonormal bases. This work is a direct generalization of noiseless BP study, and indeed, when the noise power is reduced to zero, we obtain the known results of the noiseless BP.
Donoho,
D.,
and
M.
Elad,
"On the Stability of the Basis Pursuit in the Presence of Noise ",
Signal Processing , 86 , 511532.
Authors:
Donoho
Elad
Coders:
Donoho
Elad Last update
10/08/2012
Ranking
29
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N.A.
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