* These charts can be obtained from Creaseys Ltd., Bull Lane, Hertford. Two sizes are obtainable, the smaller designed to fit in hospital note folders and the larger double foolscap size for more accurate plotting.

Consider an underdetermined system of linear equations y ‫؍‬ Ax with known y and d ؋ n matrix A. We seek the nonnegative x with the fewest nonzeros satisfying y ‫؍‬ Ax. In general, this problem is NP-hard. However, for many matrices A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be found by linear programming. We explain this by the theory of convex polytopes. Let aj denote the jth column of A, 1 < j Յ n, let a0 ‫؍‬ 0 and P denote the convex hull of the aj. We say the polytope P is outwardly k-neighborly if every subset of k vertices not including 0 spans a face of P. We show that outward k-neighborliness is 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. We also consider weak neighborliness, where the overwhelming majority of k-sets of a js not containing 0 span a face of P. This implies that most nonnegative vectors x with k nonzeros are uniquely recoverable from y ‫؍‬ Ax by linear programming. Numerous corollaries follow by invoking neighborliness results. For example, for most large n by 2n underdetermined systems having a solution with fewer nonzeros than roughly half the number of equations, the sparsest solution can be found by linear programming.neighborly polytopes ͉ cyclic polytopes ͉ combinatorial optimization ͉ convex hull of Gaussian samples ͉ positivity constraints in ill-posed problems

Let A be a d ؋ n matrix and T ‫؍‬ T n؊1 be the standard simplex in R n . Suppose that d and n are both large and comparable: d Ϸ ␦n, ␦ ʦ (0, 1). We count the faces of the projected simplex AT when the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of R n . We derive N(␦) > 0 with the property that, for any < N(␦), with overwhelming probability for large d, the number of k-dimensional 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 also study a weaker notion of neighborliness where the numbers of k-dimensional faces f k(P) > f k(T)(1 ؊ ). Vershik and Sporyshev previously showed existence of a threshold VS(␦) > 0 at which phase transition occurs in k͞d. We compute and display VS and compare with N. Corollaries are as follows. (1) neighborly polytopes ͉ convex hull of Gaussian sample ͉ underdetermined systems of linear equations ͉ uniformly distributed random projections ͉ phase transitions

We review connections between phase transitions in high-dimensional combinatorial geometry and phase transitions occurring in modern high-dimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt breakdown of linear model selection, robust data fitting or compressed sensing reconstructions, when the complexity of the model or the number of outliers increases beyond a threshold. In combinatorial geometry, these transitions appear as abrupt changes in the properties of face counts of convex polytopes when the dimensions are varied. The thresholds in these very different problems appear in the same critical locations after appropriate calibration of variables. These thresholds are important in each subject area: for linear modelling, they place hard limits on the degree to which the now ubiquitous high-throughput data analysis can be successful; for robustness, they place hard limits on the degree to which standard robust fitting methods can tolerate outliers before breaking down; for compressed sensing, they define the sharp boundary of the undersampling/sparsity trade-off curve in undersampling theorems. Existing derivations of phase transitions in combinatorial geometry assume that the underlying matrices have independent and identically distributed Gaussian elements. In applications, however, it often seems that Gaussianity is not required. We conducted an extensive computational experiment and formal inferential analysis to test the hypothesis that these phase transitions are universal across a range of underlying matrix ensembles. We ran millions of linear programs using random matrices spanning several matrix ensembles and problem sizes; visually, the empirical phase transitions do not depend on the ensemble, and they agree extremely well with the asymptotic theory assuming Gaussianity. Careful statistical analysis reveals discrepancies that can be explained as transient terms, decaying with problem size. The experimental results are thus consistent with an asymptotic large-n universality across matrix ensembles; finite-sample universality can be rejected.

Let Q = Q N Q = Q_N denote either the N N -dimensional cross-polytope C N C^N or the N − 1 N-1 -dimensional simplex T N − 1 T^{N-1} . Let A = A n , N A = A_{n,N} denote a random orthogonal projector A : R N ↦ b R n A: \mathbf {R}^{N} \mapsto bR^n . We compare the number of faces f k ( A Q ) f_k(AQ) of the projected polytope A Q AQ to the number of faces of f k ( Q ) f_k(Q) of the original polytope Q Q . We concentrate on the case where n n and N N are both large, but n n is much smaller than N N ; in this case the projection dramatically lowers dimension. We consider sequences of triples ( k , n , N ) (k,n,N) where N = N n N = N_n is not exponentially larger than n n . We identify thresholds of the form c o n s t ⋅ n log ⁡ ( n / N ) const \cdot n \log (n/N) where the relationship of f k ( A Q ) f_k(AQ) and f k ( Q ) f_k(Q) changes abruptly. These properties of polytopes have significant implications for neighborliness of convex hulls of Gaussian point clouds, for efficient sparse solution of underdetermined linear systems, for efficient decoding of random error correcting codes and for determining the allowable rate of undersampling in the theory of compressed sensing. The thresholds are characterized precisely using tools from polytope theory, convex integral geometry, and large deviations. Asymptotics developed for these thresholds yield the following, for fixed ϵ > 0 \epsilon > 0 . With probability tending to 1 as n n , N N tend to infinity: (1a) for k > ( 1 − ϵ ) ⋅ n [ 2 e ln ⁡ ( N / n ) ] − 1 k > (1-\epsilon ) \cdot n [2e\ln (N/n)]^{-1} we have f k ( A Q ) = f k ( Q ) f_k(AQ) = f_k(Q) , (1b) for k > ( 1 + ϵ ) ⋅ n [ 2 e ln ⁡ ( N / n ) ] − 1 k > (1 +\epsilon ) \cdot n [2e\ln (N/n)]^{-1} we have f k ( A Q ) > f k ( Q ) f_k(AQ) > f_k(Q) , with E {\mathcal E} denoting expectation, (2a) for k > ( 1 − ϵ ) ⋅ n [ 2 ln ⁡ ( N / n ) ] − 1 k > (1-\epsilon ) \cdot n [2\ln (N/n)]^{-1} we have E f k ( A Q ) > ( 1 − ϵ ) f k ( Q ) {\mathcal E} f_k(AQ) > (1-\epsilon ) f_k(Q) , (2b) for k > ( 1 + ϵ ) ⋅ n [ 2 ln ⁡ ( N / n ) ] − 1 k > (1 +\epsilon ) \cdot n [2\ln (N/n)]^{-1} we have E f k ( A Q ) > ϵ f k ( Q ) {\mathcal E} f_k(AQ) > \epsilon f_k(Q) . These asymptotically sharp transitions in the behavior of face numbers as k k varies relative to n log ⁡ ( N / n ) n \log (N/n) are proven, interpreted, and related to the above-mentioned applications.

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Undersampling theorems state that we may gather far fewer samples than the usual sampling theorem while exactly reconstructing the object of interest-provided the object in question obeys a sparsity condition, the samples measure appropriate linear combinations of signal values, and we reconstruct with a particular nonlinear procedure. While there are many ways to crudely demonstrate such undersampling phenomena, we know of only one mathematically rigorous approach which precisely quantifies the true sparsity-undersampling tradeoff curve of standard algorithms and standard compressed sensing matrices. That approach, based on combinatorial geometry, predicts the exact location in sparsity-undersampling domain where standard algorithms exhibitphase transitionsin performance. We review the phase transition approach here and describe the broad range of cases where it applies. We also mention exceptions and state challenge problems for future research. Sample result: one can efficiently reconstruct a k-sparse signal of length N from n measurements, provided n ?? 2k ?? log(N/n), for (k,n,N) large.k ?? N.AMS 2000 subject classifications. Primary: 41A46, 52A22, 52B05, 62E20, 68P30, 94A20; Secondary: 15A52, 60F10, 68P25, 90C25, 94B20.

Abstract. Let A be an n by N real valued random matrix, and H N denote the N -dimensional hypercube. For numerous random matrix ensembles, the expected number of k-dimensional faces of the random n-dimensional zonotope AH N obeys the formula Ef k (AH N )/f k (H N ) = 1 − P N−n,N−k , where P N−n,N−k is a fair-coin-tossing probability:The formula applies, for example, where the columns of A are drawn i.i.d. from an absolutely continuous symmetric distribution. The formula exploits Wendel's Theorem [19].Let R N + denote the positive orthant; the expected number of k-faces of the random coneARThe formula applies to numerous matrix ensembles, including those with iid random columns from an absolutely continuous, centrally symmetric distribution.The probabilities P N−n,N−k change rapidly from nearly 0 to nearly 1 near k ≈ 2n − N . Consequently, there is an asymptotically sharp threshold in the behavior of face counts of the projected hypercube; thresholds known for projecting the simplex and the cross-polytope, occur at very different locations. We briefly consider face counts of the projected orthant when A does not have mean zero; these do behave similarly to those for the projected simplex. We consider non-random projectors of the orthant; the 'best possible' A is the one associated with the first n rows of the Fourier matrix.These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Most of these flow in some way from the fact that face counting is related to conditions for uniqueness of solutions of underdetermined systems of linear equations. a) A vector in R N + is called k-sparse if it has at most k nonzeros. For such a k-sparse vector x 0 , let b = Ax 0 , where A is a random matrix ensemble covered by our results. With probability 1 − P N−n,N−k the inequality-constrained system Ax = b, x ≥ 0 has x 0 as its unique nonnegative solution. This is so, even if n < N , so that the system Ax = b is underdetermined. b) A vector in the hypercube H N will be called k-simple if all entries except at most k are at the bounds 0 or 1. For such a k-simple vector x 0 , let b = Ax 0 , where A is a random matrix ensemble covered by our results. With probability 1 − P N−n,N−k the inequality-constrained system Ax = b, x ∈ H N has x 0 as its unique solution in the hypercube.