The Convex Geometry of Linear Inverse Problems
In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered are those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases such as sparse vectors (e.g., signal processing, statistics) and low-rank matrices (e.g., control, statistics), as well as several others including sums of a few permutations matrices (e.g., ranked elections, multiobject tracking), low-rank tensors (e.g., computer vision, neuroscience), orthogonal matrices (e.g., machine learning), and atomic measures (e.g., system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery, and this tradeoff is characterized via some examples. Thus this work extends the catalog of simple models (beyond sparse vectors and low-rank matrices) that can be recovered from limited linear information via tractable convex programming.
Keywords: Convex optimization; semidefinite programming; atomic norms; real algebraic geometry; Gaussian width; symmetry.
Deducing the state or structure of a system from partial, noisy measurements is a fundamental task throughout the sciences and engineering. A commonly encountered difficulty that arises in such inverse problems is the limited availability of data relative to the ambient dimension of the signal to be estimated. However many interesting signals or models in practice contain few degrees of freedom relative to their ambient dimension. For instance a small number of genes may constitute a signature for disease, very few parameters may be required to specify the correlation structure in a time series, or a sparse collection of geometric constraints might completely specify a molecular configuration. Such low-dimensional structure plays an important role in making inverse problems well-posed. In this paper we propose a unified approach to transform notions of simplicity into convex penalty functions, thus obtaining convex optimization formulations for inverse problems.
We describe a model as simple if it can be written as a nonnegative combination of a few elements from an atomic set. Concretely let be formed as follows:
where is a set of atoms that constitute simple building blocks of general signals. Here we assume that is simple so that is relatively small. For example could be the finite set of unit-norm one-sparse vectors in which case is a sparse vector, or could be the infinite set of unit-norm rank-one matrices in which case is a low-rank matrix. These two cases arise in many applications, and have received a tremendous amount of attention recently as several authors have shown that sparse vectors and low-rank matrices can be recovered from highly incomplete information [16, 26, 27, 62, 17]. However a number of other structured mathematical objects also fit the notion of simplicity described in (1). The set could be the collection of unit-norm rank-one tensors, in which case is a low-rank tensor and we are faced with the familiar challenge of low-rank tensor decomposition. Such problems arise in numerous applications in computer vision and image processing , and in neuroscience . Alternatively could be the set of permutation matrices; sums of a few permutation matrices are objects of interest in ranking  and multi-object tracking. As yet another example, could consist of measures supported at a single point so that is an atomic measure supported at just a few points. This notion of simplicity arises in problems in system identification and statistics.
In each of these examples as well as several others, a fundamental problem of interest is to recover given limited linear measurements. For instance the question of recovering a sparse function over the group of permutations (i.e., the sum of a few permutation matrices) given linear measurements in the form of partial Fourier information was investigated in the context of ranked election problems . Similar linear inverse problems arise with atomic measures in system identification, with orthogonal matrices in machine learning, and with simple models formed from several other atomic sets (see Section 2.2 for more examples). Hence we seek tractable computational tools to solve such problems. When is the collection of one-sparse vectors, a method of choice is to use the norm to induce sparse solutions. This method has seen a surge in interest in the last few years as it provides a tractable convex optimization formulation to exactly recover sparse vectors under various conditions [16, 26, 27]. More recently the nuclear norm has been proposed as an effective convex surrogate for solving rank minimization problems subject to various affine constraints [62, 17].
Motivated by the success of these methods we propose a general convex optimization framework in Section 2 in order to recover objects with structure of the form (1) from limited linear measurements. The guiding question behind our framework is: how do we take a concept of simplicity such as sparsity and derive the norm as a convex heuristic? In other words what is the natural procedure to go from the set of one-sparse vectors to the norm? We observe that the convex hull of (unit-Euclidean-norm) one-sparse vectors is the unit ball of the norm, or the cross-polytope. Similarly the convex hull of the (unit-Euclidean-norm) rank-one matrices is the nuclear norm ball; see Figure 1 for illustrations. These constructions suggest a natural generalization to other settings. Under suitable conditions the convex hull defines the unit ball of a norm, which is called the atomic norm induced by the atomic set . We can then minimize the atomic norm subject to measurement constraints, which results in a convex programming heuristic for recovering simple models given linear measurements. As an example suppose we wish to recover the sum of a few permutation matrices given linear measurements. The convex hull of the set of permutation matrices is the Birkhoff polytope of doubly stochastic matrices , and our proposal is to solve a convex program that minimizes the norm induced by this polytope. Similarly if we wish to recover an orthogonal matrix from linear measurements we would solve a spectral norm minimization problem, as the spectral norm ball is the convex hull of all orthogonal matrices. As discussed in Section 2.5 the atomic norm minimization problem is, in some sense, the best convex heuristic for recovering simple models with respect to a given atomic set.
We give general conditions for exact and robust recovery using the atomic norm heuristic. In Section 3 we provide concrete bounds on the number of generic linear measurements required for the atomic norm heuristic to succeed. This analysis is based on computing certain Gaussian widths of tangent cones with respect to the unit balls of the atomic norm . Arguments based on Gaussian width have been fruitfully applied to obtain bounds on the number of Gaussian measurements for the special case of recovering sparse vectors via norm minimization [64, 67], but computing Gaussian widths of general cones is not easy. Therefore it is important to exploit the special structure in atomic norms, while still obtaining sufficiently general results that are broadly applicable. An important theme in this paper is the connection between Gaussian widths and various notions of symmetry. Specifically by exploiting symmetry structure in certain atomic norms as well as convex duality properties, we give bounds on the number of measurements required for recovery using very general atomic norm heuristics. For example we provide precise estimates of the number of generic measurements required for exact recovery of an orthogonal matrix via spectral norm minimization, and the number of generic measurements required for exact recovery of a permutation matrix by minimizing the norm induced by the Birkhoff polytope. While these results correspond to the recovery of individual atoms from random measurements, our techniques are more generally applicable to the recovery of models formed as sums of a few atoms as well. We also give tighter bounds than those previously obtained on the number of measurements required to robustly recover sparse vectors and low-rank matrices via norm and nuclear norm minimization. In all of the cases we investigate, we find that the number of measurements required to reconstruct an object is proportional to its intrinsic dimension rather than the ambient dimension, thus confirming prior folklore. See Table 1 for a summary of these results.
|Underlying model||Convex heuristic||Gaussian measurements|
|-sparse vector in||norm|
|rank- matrix||nuclear norm|
|permutation matrix||norm induced by Birkhoff polytope|
|orthogonal matrix||spectral norm|
Although our conditions for recovery and bounds on the number of measurements hold generally, we note that it may not be possible to obtain a computable representation for the convex hull of an arbitrary set of atoms . This leads us to another important theme of this paper, which we discuss in Section 4, on the connection between algebraic structure in and the semidefinite representability of the convex hull . In particular when is an algebraic variety the convex hull can be approximated as (the projection of) a set defined by linear matrix inequalities. Thus the resulting atomic norm minimization heuristic can be solved via semidefinite programming. A second issue that arises in practice is that even with algebraic structure in the semidefinite representation of may not be computable in polynomial time, which makes the atomic norm minimization problem intractable to solve. A prominent example here is the tensor nuclear norm ball, obtained by taking the convex hull of the rank-one tensors. In order to address this problem we give a hierarchy of semidefinite relaxations using theta bodies that approximate the original (intractable) atomic norm minimization problem . We also highlight that while these semidefinite relaxations are more tractable to solve, we require more measurements for exact recovery of the underlying model than if we solve the original intractable atomic norm minimization problem. Hence there is a tradeoff between the complexity of the recovery algorithm and the number of measurements required for recovery. We illustrate this tradeoff with the cut polytope and its relaxations.
Outline Section 2 describes the construction of the atomic norm, gives several examples of applications in which these norms may be useful to recover simple models, and provides general conditions for recovery by minimizing the atomic norm. In Section 3 we investigate the number of generic measurements for exact or robust recovery using atomic norm minimization, and give estimates in a number of settings by analyzing the Gaussian width of certain tangent cones. We address the problem of semidefinite representability and tractable relaxations of the atomic norm in Section 4. Section 5 describes some algorithmic issues as well as a few simulation results, and we conclude with a discussion and open questions in Section 6.
2 Atomic Norms and Convex Geometry
In this section we describe the construction of an atomic norm from a collection of simple atoms. In addition we give several examples of atomic norms, and discuss their properties in the context of solving ill-posed linear inverse problems. We denote the Euclidean norm by .
Let be a collection of atoms that is a compact subset of . We will assume throughout this paper that no element lies in the convex hull of the other elements , i.e., the elements of are the extreme points of . Let denote the gauge of :
Note that the gauge is always a convex, extended-real valued function for any set . By convention this function evaluates to if does not lie in the affine hull of . We will assume without loss of generality that the centroid of is at the origin, as this can be achieved by appropriate recentering. With this assumption the gauge function can be rewritten as :
If is centrally symmetric about the origin (i.e., if and only if ) we have that is a norm, which we call the atomic norm induced by . The support function of is given as:
If is a norm the support function is the dual norm of this atomic norm. From this definition we see that the unit ball of is equal to . In many examples of interest the set is not centrally symmetric, so that the gauge function does not define a norm. However our analysis is based on the underlying convex geometry of , and our results are applicable even if does not define a norm. Therefore, with an abuse of terminology we generally refer to as the atomic norm of the set even if is not a norm. We note that the duality characterization between (2) and (3) when is a norm is in fact applicable even in infinite-dimensional Banach spaces by Bonsall’s atomic decomposition theorem , but our focus is on the finite-dimensional case in this work. We investigate in greater detail the issues of representability and efficient approximation of these atomic norms in Section 4.
Equipped with a convex penalty function given a set of atoms, we propose a convex optimization method to recover a “simple” model given limited linear measurements. Specifically suppose that is formed according to (1) from a set of atoms . Further suppose that we have a known linear map , and we have linear information about as follows:
The goal is to reconstruct given . We consider the following convex formulation to accomplish this task:
When is the set of one-sparse atoms this problem reduces to standard norm minimization. Similarly when is the set of rank-one matrices this problem reduces to nuclear norm minimization. More generally if the atomic norm is tractable to evaluate, then (5) potentially offers an efficient convex programming formulation for reconstructing from the limited information . The dual problem of (5) is given as follows:
Here denotes the adjoint (or transpose) of the linear measurement map .
The convex formulation (5) can be suitably modified in case we only have access to inaccurate, noisy information. Specifically suppose that we have noisy measurements where represents the noise term. A natural convex formulation is one in which the constraint of (5) is replaced by the relaxed constraint , where is an upper bound on the size of the noise :
We say that we have exact recovery in the noise-free case if in (5), and robust recovery in the noisy case if the error is small in (7). In Section 2.4 and Section 3 we give conditions under which the atomic norm heuristics (5) and (7) recover exactly or approximately. Atomic norms have found fruitful applications in problems in approximation theory of various function classes [58, 43, 3, 24]. However this prior body of work was concerned with infinite-dimensional Banach spaces, and none of these references consider nor provide recovery guarantees that are applicable in our setting.
Next we provide several examples of atomic norms that can be viewed as special cases of the construction above. These norms are obtained by convexifying atomic sets that are of interest in various applications.
Sparse vectors. The problem of recovering sparse vectors from limited measurements has received a great deal of attention, with applications in many problem domains. In this case the atomic set can be viewed as the set of unit-norm one-sparse vectors , and -sparse vectors in can be constructed using a linear combination of elements of the atomic set. In this case it is easily seen that the convex hull is given by the cross-polytope (i.e., the unit ball of the norm), and the atomic norm corresponds to the norm in .
Low-rank matrices. Recovering low-rank matrices from limited information is also a problem that has received considerable attention as it finds applications in problems in statistics, control, and machine learning. The atomic set here can be viewed as the set of rank-one matrices of unit-Euclidean-norm. The convex hull is the nuclear norm ball of matrices in which the sum of the singular values is less than or equal to one.
Sparse and low-rank matrices. The problem of recovering a sparse matrix and a low-rank matrix given information about their sum arises in a number of model selection and system identification settings. The corresponding atomic norm is constructed by taking the convex hull of an atomic set obtained via the union of rank-one matrices and (suitably scaled) one-sparse matrices. This norm can also be viewed as the infimal convolution of the norm and the nuclear norm, and its properties have been explored in [19, 14].
Permutation matrices. A problem of interest in a ranking context  or an object tracking context is that of recovering permutation matrices from partial information. Suppose that a small number of rankings of candidates is preferred by a population. Such preferences can be modeled as the sum of a few permutation matrices, with each permutation corresponding to a particular ranking. By conducting surveys of the population one can obtain partial linear information of these preferred rankings. The set here is the collection of permutation matrices (consisting of elements), and the convex hull is the Birkhoff polytope or the set of doubly stochastic matrices . The centroid of the Birkhoff polytope is the matrix , so it needs to be recentered appropriately. We mention here recent work by Jagabathula and Shah  on recovering a sparse function over the symmetric group (i.e., the sum of a few permutation matrices) given partial Fourier information; although the algorithm proposed in  is tractable it is not based on convex optimization.
Binary vectors. In integer programming one is often interested in recovering vectors in which the entries take on values of . Suppose that there exists such a sign-vector, and we wish to recover this vector given linear measurements. This corresponds to a version of the multi-knapsack problem . In this case is the set of all sign-vectors, and the convex hull is the hypercube or the unit ball of the norm. The image of this hypercube under a linear map is also referred to as a zonotope .
Vectors from lists. Suppose there is an unknown vector , and that we are given the entries of this vector without any information about the locations of these entries. For example if , then we are only given the list of numbers without their positions in . Further suppose that we have access to a few linear measurements of . Can we recover by solving a convex program? Such a problem is of interest in recovering partial rankings of elements of a set. An extreme case is one in which we only have two preferences for rankings, i.e., a vector in composed only of one’s and two’s, which reduces to a special case of the problem above of recovering binary vectors (in which the number of entries of each sign is fixed). For this problem the set is the set of all permutations of (which we know since we have the list of numbers that compose ), and the convex hull is the permutahedron [73, 65]. As with the Birkhoff polytope, the permutahedron also needs to be recentered about the point .
Matrices constrained by eigenvalues. This problem is in a sense the non-commutative analog of the one above. Suppose that we are given the eigenvalues of a symmetric matrix, but no information about the eigenvectors. Can we recover such a matrix given some additional linear measurements? In this case the set is the set of all symmetric matrices with eigenvalues , and the convex hull is given by the Schur-Horn orbitope .
Orthogonal matrices. In many applications matrix variables are constrained to be orthogonal, which is a non-convex constraint and may lead to computational difficulties. We consider one such simple setting in which we wish to recover an orthogonal matrix given limited information in the form of linear measurements. In this example the set is the set of orthogonal matrices, and is the spectral norm ball.
Measures. Recovering a measure given its moments is another question of interest that arises in system identification and statistics. Suppose one is given access to a linear combination of moments of an atomically supported measure. How can we reconstruct the support of the measure? The set here is the moment curve, and its convex hull goes by several names including the Caratheodory orbitope . Discretized versions of this problem correspond to the set being a finite number of points on the moment curve; the convex hull is then a cyclic polytope .
Cut matrices. In some problems one may wish to recover low-rank matrices in which the entries are constrained to take on values of . Such matrices can be used to model basic user preferences, and are of interest in problems such as collaborative filtering . The set of atoms could be the set of rank-one signed matrices, i.e., matrices of the form with the entries of being . The convex hull of such matrices is the cut polytope . An interesting issue that arises here is that the cut polytope is in general intractable to characterize. However there exist several well-known tractable semidefinite relaxations to this polytope [25, 36], and one can employ these in constructing efficient convex programs for recovering cut matrices. We discuss this point in greater detail in Section 4.3.
Low-rank tensors. Low-rank tensor decompositions play an important role in numerous applications throughout signal processing and machine learning . Developing computational tools to recover low-rank tensors is therefore of great interest. In principle we could solve a tensor nuclear norm minimization problem, in which the tensor nuclear norm ball is obtained by taking the convex hull of rank-one tensors. A computational challenge here is that the tensor nuclear norm is in general intractable to compute; in order to address this problem we discuss further convex relaxations to the tensor nuclear norm using theta bodies in Section 4. A number of additional technical issues also arise with low-rank tensors including the non-existence in general of a singular value decomposition analogous to that for matrices , and the difference between the rank of a tensor and its border rank .
Nonorthogonal factor analysis. Suppose that a data matrix admits a factorization . The matrix nuclear norm heuristic will find a factorization into orthogonal factors in which the columns of and rows of are mutually orthogonal. However if a priori information is available about the factors, precision and recall could be improved by enforcing such priors. These priors may sacrifice orthogonality, but the factors might better conform with assumptions about how the data are generated. For instance in some applications one might know in advance that the factors should only take on a discrete set of values . In this case, we might try to fit a sum of rank-one matrices that are bounded in norm rather than in norm. Another prior that commonly arises in practice is that the factors are non-negative (i.e., in non-negative matrix factorization). These and other priors on the basic rank-one summands induce different norms on low-rank models than the standard nuclear norm , and may be better suited to specific applications.
2.3 Background on Tangent and Normal Cones
In order to properly state our results, we recall some basic concepts from convex analysis. A convex set is a cone if it is closed under positive linear combinations. The polar of a cone is the cone
Given some nonzero we define the tangent cone at with respect to the scaled unit ball as
The cone is equal to the set of descent directions of the atomic norm at the point , i.e., the set of all directions such that the directional derivative is negative.
The normal cone at with respect to the scaled unit ball is defined to be the set of all directions that form obtuse angles with every descent direction of the atomic norm at the point :
The normal cone is equal to the set of all normals of hyperplanes given by normal vectors that support the scaled unit ball at . Observe that the polar cone of the tangent cone is the normal cone and vice-versa. Moreover we have the basic characterization that the normal cone is the conic hull of the subdifferential of the atomic norm at .
2.4 Recovery Condition
The following result gives a characterization of the favorable underlying geometry required for exact recovery. Let denote the nullspace of the operator .
We have that is the unique optimal solution of (5) if and only if .
Proposition 2.1 asserts that the atomic norm heuristic succeeds if the nullspace of the sampling operator does not intersect the tangent cone at . In Section 3 we provide a characterization of tangent cones that determines the number of Gaussian measurements required to guarantee such an empty intersection.
A tightening of this empty intersection condition can also be used to address the noisy approximation problem. The following proposition characterizes when can be well-approximated using the convex program (7).
Suppose that we are given noisy measurements where , and . Let denote an optimal solution of (7). Further suppose for all that we have . Then .
The set of descent directions at with respect to the atomic norm ball is given by the tangent cone . The error vector lies in because is a minimal atomic norm solution, and hence . It follows by the triangle inequality that
By assumption we have that
which allows us to conclude that . ∎
Therefore, we need only concern ourselves with estimating the minimum value of for non-zero . We denote this quantity as the minimum gain of the measurement operator restricted to the cone . In particular if this minimum gain is bounded away from zero, then the atomic norm heuristic also provides robust recovery when we have access to noisy linear measurements of . Minimum gain conditions have been employed in recent recovery results via -norm minimization, block-sparse vector recovery, and low-rank matrix reconstruction [18, 8, 70, 53]. All of these results rely heavily on strong decomposability conditions of the norm and the matrix nuclear norm. However, there are several examples of atomic norms (for instance, the norm and the norm induced by the Birkhoff polytope) as specified in Section 2.2 that do not satisfy such decomposability conditions. As we ll see in the sequel the more geometric viewpoint adopted in this paper provides a fruitful framework in which to analyze the recovery properties of general atomic norms.
2.5 Why Atomic Norm?
The atomic norm induced by a set possesses a number of favorable properties that are useful for recovering “simple” models from limited linear measurements. The key point to note from Section 2.4 is that the smaller the tangent cone at a point with respect to , the easier it is to satisfy the empty-intersection condition of Proposition 2.1.
Based on this observation it is desirable that points in with smaller tangent cones correspond to simpler models, while points in with larger tangent cones generally correspond to more complicated models. The construction of by taking the convex hull of ensures that this is the case. The extreme points of correspond to the simplest models, i.e., those models formed from a single element of . Further the low-dimensional faces of consist of those elements that are obtained by taking linear combinations of a few basic atoms from . These are precisely the properties desired as points lying in these low-dimensional faces of have smaller tangent cones than those lying on larger faces.
We also note that the atomic norm is, in some sense, the best possible convex heuristic for recovering simple models. Any reasonable heuristic penalty function should be constant on the set of atoms . This ensures that no atom is preferred over any other. Under this assumption, we must have that for any , must be a descent direction for all . The best convex penalty function is one in which the cones of descent directions at are as small as possible. This is because, as described above, smaller cones are more likely to satisfy the empty intersection condition required for exact recovery. Since the tangent cone at with respect to is precisely the conic hull of for , the atomic norm is the best convex heuristic for recovering models where simplicity is dictated by the set .
Our reasons for proposing the atomic norm as a useful convex heuristic are quite different from previous justifications of the norm and the nuclear norm. In particular let denote the cardinality function that counts the number of nonzero entries of a vector. Then the norm is the convex envelope of restricted to the unit ball of the norm, i.e., the best convex underestimator of restricted to vectors in the -norm ball. This view of the norm in relation to the function is often given as a justification for its effectiveness in recovering sparse vectors. However if we consider the convex envelope of restricted to the Euclidean norm ball, then we obtain a very different convex function than the norm! With more general atomic sets, it may not be clear a priori what the bounding set should be in deriving the convex envelope. In contrast the viewpoint adopted in this paper leads to a natural, unambiguous construction of the norm and other general atomic norms. Further as explained above it is the favorable facial structure of the atomic norm ball that makes the atomic norm a suitable convex heuristic to recover simple models, and this connection is transparent in the definition of the atomic norm.
3 Recovery from Generic Measurements
We consider the question of using the convex program (5) to recover “simple” models formed according to (1) from a generic measurement operator or map . Specifically, we wish to compute estimates on the number of measurements so that we have exact recovery using (5) for most operators comprising of measurements. That is, the measure of -measurement operators for which recovery fails using (5) must be exponentially small. In order to conduct such an analysis we study random Gaussian maps whose entries are independent and identically distributed Gaussians. These measurement operators have a nullspace that is uniformly distributed among the set of all -dimensional subspaces in . In particular we analyze when such operators satisfy the conditions of Proposition 2.1 and Proposition 2.2 for exact recovery.
3.1 Recovery Conditions based on Gaussian Width
Proposition 2.1 requires that the nullspace of the measurement operator must miss the tangent cone . Gordon  gave a solution to the problem of characterizing the probability that a random subspace (of some fixed dimension) distributed uniformly misses a cone. We begin by defining the Gaussian width of a set, which plays a key role in Gordon’s analysis.
The Gaussian width of a set is defined as:
where is a vector of independent zero-mean unit-variance Gaussians.
Gordon characterized the likelihood that a random subspace misses a cone purely in terms of the dimension of the subspace and the Gaussian width , where is the unit sphere. Before describing Gordon’s result formally, we introduce some notation. Let denote the expected length of a -dimensional Gaussian random vector. By elementary integration, we have that . Further by induction one can show that is tightly bounded as .
The main idea underlying Gordon’s theorem is a bound on the minimum gain of an operator restricted to a set. Specifically, recall that is the condition required for recovery by Proposition 2.1. Thus if we have that the minimum gain of restricted to vectors in the set is bounded away from zero, then it is clear that . We refer to such minimum gains restricted to a subset of the sphere as restricted minimum singular values, and the following theorem of Gordon gives a bound these quantities:
Theorem 3.2 (Cor. 1.2, ).
Let be a closed subset of . Let be a random map with i.i.d. zero-mean Gaussian entries having variance one. Then
Theorem 3.2 allows us to characterize exact recovery in the noise-free case using the convex program (5), and robust recovery in the noisy case using the convex program (7). Specifically, we consider the number of measurements required for exact or robust recovery when the measurement map consists of i.i.d. zero-mean Gaussian entries having variance . The normalization of the variance ensures that the columns of are approximately unit-norm, and is necessary in order to properly define a signal-to-noise ratio. The following corollary summarizes the main results of interest in our setting:
Let be a random map with i.i.d. zero-mean Gaussian entries having variance . Further let denote the spherical part of the tangent cone .
The two results are simple consequences of Theorem 3.2 and a concentration of measure argument. Recall that for an function with Lipschitz constant and a random Gaussian vector, , with mean zero and identity variance
provided that .
The first part now follows by setting in (14). The concentration inequality is valid provided that . To verify this, note
Here, both inequalities use the fact that .
For the second part, we have from (14) that
for all with high probability if . In this case, we can apply Proposition 2.2 to conclude that . To verify that concentration of measure can be applied is more or less the same as in the proof of Part 1. First, note that under the assumptions of the theorem
as for . Using this fact, we then have
as desired. ∎
Gordon’s theorem thus provides a simple characterization of the number of measurements required for reconstruction with the atomic norm. Indeed the Gaussian width of is the only quantity that we need to compute in order to obtain bounds for both exact and robust recovery. Unfortunately it is in general not easy to compute Gaussian widths. Rudelson and Vershynin  have worked out Gaussian widths for the special case of tangent cones at sparse vectors on the boundary of the ball, and derived results for sparse vector recovery using minimization that improve upon previous results. In the next section we give various well-known properties of the Gaussian width that are useful in computations. In Section 3.3 we discuss a new approach to width computations that gives near-optimal recovery bounds in a variety of settings.
3.2 Properties of Gaussian Width
The Gaussian width has deep connections to convex geometry. Since the length and direction of a Gaussian random vector are independent, one can verify that for
where the integral is with respect to Haar measure on and is known as the mean width of . The mean width measures the average length of along unit directions in and is one of the fundamental intrinsic volumes of a body studied in combinatorial geometry . Any continuous valuation that is invariant under rigid motions and homogeneous of degree 1 is a multiple of the mean width and hence a multiple of the Gaussian width. We can use this connection with convex geometry to underscore several properties of the Gaussian width that are useful for computation.
The Gaussian width of a body is invariant under translations and unitary transformations. Moreover, it is homogeneous in the sense that = for . The width is also monotonic. If , then it is clear from the definition of the Gaussian width that
Less obvious, the width is modular in the sense that if and are convex bodies with convex, we also have
This equality follows from the fact that is a valuation . Also note that if we have a set , then the Gaussian width of is equal to the Gaussian width of the convex hull of :
This result follows from the basic fact in convex analysis that the maximum of a convex function over a convex set is achieved at an extreme point of the convex set.
If is a subspace in , then we have that
which follows from standard results on random Gaussians. This result also agrees with the intuition that a random Gaussian map misses a -dimensional subspace with high probability as long as . Finally, if a cone is such that , where is a -dimensional cone, is a -dimensional cone that is orthogonal to , and denotes the direct sum operation, then the width can be decomposed as follows:
These observations are useful in a variety of situations. For example a width computation that frequently arises is one in which as described above, with being a -dimensional subspace. It follows that the width of is bounded as
Let be an arbitrary compact subset of . The covering number of in the Euclidean norm at resolution is the smallest number, , such that Euclidean balls of radius cover .
Theorem 3.5 (Dudley’s Inequality).
Let be an arbitrary compact subset of , and let be a random vector with i.i.d. zero-mean, unit-variance Gaussian entries. Then
We note here that a weak converse to Dudley’s inequality can be obtained via Sudakov’s Minoration  by using the covering number for just a single scale. Specifically, we have the following lower bound on the Gaussian width of a compact subset for any :
Here is some universal constant.
Although Dudley’s inequality can be applied quite generally, estimating covering numbers is difficult in most instances. There are a few simple characterizations available for spheres and Sobolev spaces, and some tractable arguments based on Maurey’s empirical method . However it is not evident how to compute these numbers for general convex cones. Also, in order to apply Dudley’s inequality we need to estimate the covering number at all scales. Further Dudley’s inequality can be quite loose in its estimates, and it often introduces extraneous polylogarithmic factors. In the next section we describe a new mechanism for estimating Gaussian widths, which provides near-optimal guarantees for recovery of sparse vectors and low-rank matrices, as well as for several of the recovery problems discussed in Section 3.4.
3.3 New Results on Gaussian Width
We now present a framework for computing Gaussian widths by bounding the Gaussian width of a cone via the distance to the dual cone. To be fully general let be a non-empty convex cone in , and let denote the polar of . We can then upper bound the Gaussian width of any cone in terms of the polar cone :
Let be any non-empty convex cone in , and let be a random Gaussian vector. Then we have the following bound:
where here denotes the Euclidean distance between a point and a set.
The proof is given in Appendix A, and it follows from an appeal to convex duality. Proposition 3.6 is more or less a restatement of the fact that the support function of a convex cone is equal to the distance to its polar cone. As it is the square of the Gaussian width that is of interest to us (see Corollary 3.3), it is often useful to apply Jensen’s inequality to make the following approximation:
The inspiration for our characterization in Proposition 3.6 of the width of a cone in terms of the expected distance to its dual came from the work of Stojnic , who used linear programming duality to construct Gaussian-width-based estimates for analyzing recovery in sparse reconstruction problems. Specifically, Stojnic’s relatively simple approach recovered well-known phase transitions in sparse signal recovery , and also generalized to block sparse signals and other forms of structured sparsity.
This new dual characterization yields a number of useful bounds on the Gaussian width, which we describe here. In the following section we use these bounds to derive new recovery results. The first result is a bound on the Gaussian width of a cone in terms of the Gaussian width of its polar.
Let be a non-empty closed, convex cone. Then we have that
In many recovery problems one is interested in computing the width of a self-dual cone. For such cones the following corollary to Lemma 3.7 gives a simple solution:
Let be a self-dual cone, i.e., . Then we have that
The proof follows directly from Lemma 3.7 as . ∎
Our next bound for the width of a cone is based on the volume of its polar . The volume of a measurable subset of the sphere is the fraction of the sphere covered by the subset. Thus it is a quantity between zero and one.
Theorem 3.9 (Gaussian width from volume of the polar).
Let be any closed, convex, solid cone, and suppose that its polar is such that has a volume of . Then for we have that
The proof of this theorem is given in Appendix B. The main property that we appeal to in the proof is Gaussian isoperimetry. In particular there is a formal sense in which a spherical cap111A spherical cap is a subset of the sphere obtained by intersecting the sphere with a halfspace. is the “extremal case” among all subsets of the sphere with a given volume . Other than this observation the proof mainly involves a sequence of integral calculations.
Note that if we are given a specification of a cone in terms of a membership oracle, it is possible to efficiently obtain good numerical estimates of the volume of . Moreover, simple symmetry arguments often give relatively accurate estimates of these volumes. Such estimates can then be plugged into Theorem 3.9 to yield bounds on the width.
3.4 New Recovery Bounds
We use the bounds derived in the last section to obtain new recovery results. First using the dual characterization of the Gaussian width in Proposition 3.6, we are able to obtain sharp bounds on the number of measurements required for recovering sparse vectors and low-rank matrices from random Gaussian measurements using convex optimization (i.e., -norm and nuclear norm minimization).
Let be an -sparse vector. Letting denote the set of unit-Euclidean-norm one-sparse vectors, we have that
Thus, random Gaussian measurements suffice to recover via norm minimization with high probability.
Let be an rank- matrix with . Letting denote the set of unit-Euclidean-norm rank-one matrices, we have that
Thus random Gaussian measurements suffice to recover via nuclear norm minimization with high probability.
The proofs of these propositions are given in Appendix C. The number of measurements required by these bounds is on the same order as previously known results. In the case of sparse vectors, previous results getting were asymptotic . Our bounds, in contrast, hold with high probability in finite dimensions. In the case of low-rank matrices, our bound provides considerably sharper constants than those previously derived (as in, for example ). We also note that we have robust recovery at these thresholds. Further these results do not require explicit recourse to any type of restricted isometry property , and the proofs are simple and based on elementary integrals.
Next we obtain a set of recovery results by appealing to Corollary 3.8 on the width of a self-dual cone. These examples correspond to the recovery of individual atoms (i.e., the extreme points of the set ), although the same machinery is applicable in principle to estimate the number of measurements required to recover models formed as sums of a few atoms (i.e., points lying on low-dimensional faces of ). We first obtain a well-known result on the number of measurements required for recovering sign-vectors via norm minimization.
Let be the set of sign-vectors in . Suppose is a vector formed as a convex combination of sign-vectors in such that lies on a -face of the -norm unit ball. Then we have that
Thus random Gaussian measurements suffice to recover via -norm minimization with high probability.
This result agrees with previously computed bounds in [51, 30], which relied on a more complicated combinatorial argument. Next we compute the number of measurements required to recover orthogonal matrices via spectral-norm minimization (see Section 2.2). Let denote the group of orthogonal matrices, viewed as a subgroup of the set of nonsingular matrices in .
Let be an orthogonal matrix, and let be the set of all orthogonal matrices. Then we have that
Thus random Gaussian measurements suffice to recover via spectral-norm minimization with high probability.
Due to the symmetry of the orthogonal group, it suffices to consider the tangent cone at the identity matrix with respect to the spectral norm ball. Recall that the spectral norm ball is the convex hull of the orthogonal matrices. Therefore the tangent space at the identity matrix with respect to the orthogonal group is a subset of the tangent cone . It is well-known that this tangent space is the Lie Algebra of all skew-symmetric matrices. Thus we only need to compute the component of that lies in the subspace of symmetric matrices:
We note that the number of degrees of freedom in an orthogonal matrix (i.e., the dimension of the manifold of orthogonal matrices) is . Proposition 3.12 and Proposition 3.13 point to the importance of obtaining recovery bounds with sharp constants. Larger constants in either result would imply that the number of measurements required exceeds the ambient dimension of the underlying . In these and many other cases of interest Gaussian width arguments not only give order-optimal recovery results, but also provide precise constants that result in sharp recovery thresholds.
Finally we give a third set of recovery results that appeal to the Gaussian width bound of Theorem 3.9. The following measurement bound applies to cases when is a symmetric polytope (roughly speaking, all the vertices are “equivalent”), and is a simple corollary of Theorem 3.9.
Suppose that the set is a finite collection of points, with the convex hull being a vertex-transitive polytope  whose vertices are the points in . Using the convex program (5) we have that random Gaussian measurements suffice, with high probability, for exact recovery of a point in , i.e., a vertex of .
We recall the basic fact from convex analysis that the normal cones at the vertices of a convex polytope in provide a partitioning of . As is a vertex-transitive polytope, the normal cone at a vertex covers fraction of . Applying Theorem 3.9, we have the desired result. ∎
Clearly we require the number of vertices to be bounded as , so that the estimate of the number of measurements is not vacuously true. This result has useful consequences in settings in which is a combinatorial polytope, as such polytopes are often vertex-transitive. We have the following example on the number of measurements required to recover permutation matrices222While Proposition 3.15 follows as a consequence of the general result in Corollary 3.14, one can remove the constant factor in the statement of Proposition 3.15 by carrying out a more refined analysis of the Birkhoff polytope.:
Let be a permutation matrix, and let be the set of all permutation matrices. Then random Gaussian measurements suffice, with high probability, to recover by solving the optimization problem (5), which minimizes the norm induced by the Birkhoff polytope of doubly stochastic matrices.
This result follows from Corollary 3.14 by noting that there are permutation matrices of size . ∎
4 Representability and Algebraic Geometry of Atomic Norms
All of our discussion thus far has focussed on arbitrary atomic sets . As seen in Section 2 the geometry of the convex hull completely determines conditions under which exact recovery is possible using the convex program (5). In this section we address the question of computing atomic norms for general sets of atoms. These issues are critical in order to be able to solve the convex optimization problem (5). Although the convex hull is always a mathematically well-defined object, testing membership in this set is in general undecidable (for example, if is a fractal). Further, even if these convex hulls are computable they may not admit efficient representations. For example if is the set of rank-one signed matrices (see Section 2.2), the corresponding convex hull is the cut polytope for which there is no known tractable characterization. Consequently, one may have to resort to efficiently computable approximations of . The tradeoff in using such approximations in our atomic norm minimization framework is that we require more measurements for robust recovery. This section is devoted to providing a better understanding of these issues.
4.1 Role of Algebraic Structure
In order to obtain exact or approximate representations (analogous to the cases of the norm and the nuclear norm) it is important to identify properties of the atomic set that can be exploited computationally. We focus on cases in which the set has algebraic structure. Specifically let the ring of multivariate polynomials in variables be denoted by . We then consider real algebraic varieties :
A real algebraic variety is the set of real solutions of a system of polynomial equations:
where is a finite collection of polynomials in .
Indeed all of the atomic sets considered in this paper are examples of algebraic varieties. Algebraic varieties have the remarkable property that (the closure of) their convex hull can be arbitrarily well-approximated in a constructive manner as (the projection of) a set defined by linear matrix inequality constraints [38, 57]. A potential complication may arise, however, if these semidefinite representations are intractable to compute in polynomial time. In such cases it is possible to approximate the convex hulls via a hierarchy of tractable semidefinite relaxations. We describe these results in more detail in Section 4.2. Therefore the atomic norm minimization problems such as (7) arising in such situations can be solved exactly or approximately via semidefinite programming.
Algebraic structure also plays a second important role in atomic norm minimization problems. If an atomic norm is intractable to compute, we may approximate it via a more tractable norm . However not every approximation of the atomic norm is equally good for solving inverse problems. As illustrated in Figure 2 we can construct approximations of the ball that are tight in a metric sense, with , but where the tangent cones at sparse vectors in the new norm are halfspaces. In such a case, the number of measurements required to recover the sparse vector ends up being on the same order as the ambient dimension. (Note that the -norm is in fact tractable to compute; we simply use it here for illustrative purposes.) The key property that we seek in approximations to an atomic norm is that they preserve algebraic structure such as the vertices/extreme points and more generally the low-dimensional faces of the . As discussed in Section 2.5 points on such low-dimensional faces correspond to simple models, and algebraic-structure preserving approximations ensure that the tangent cones at simple models with respect to the approximations are not too much larger than the corresponding tangent cones with respect to the original atomic norms (see Section 4.3 for a concrete example).
4.2 Semidefinite Relaxations using Theta Bodies
In this section we give a family of semidefinite relaxations to the atomic norm minimization problem whenever the atomic set has algebraic structure. To begin with if we approximate the atomic norm by another atomic norm defined using a larger collection of atoms , it is clear that
Consequently outer approximations of the atomic set give rise to approximate norms that provide lower bounds on the optimal value of the problem (5).
In order to provide such lower bounds on the optimal value of (5), we discuss semidefinite relaxations of the convex hull . All our discussion here is based on results described in  for semidefinite relaxations of convex hulls of algebraic varieties using theta bodies. We only give a brief review of the relevant constructions, and refer the reader to the vast literature on this subject for more details (see [38, 57] and the references therein). For subsequent reference in this section, we recall the definition of a polynomial ideal [9, 40]:
A polynomial ideal is a subset of the ring of polynomials that contains the zero polynomial (the polynomial that is identically zero), is closed under addition, and has the property that implies that .
To begin with we note that a sum-of-squares (SOS) polynomial in is a polynomial that can be written as the (finite) sum of squares of other polynomials in . Verifying the nonnegativity of a multivariate polynomial is intractable in general, and therefore SOS polynomials play an important role in real algebraic geometry as an SOS polynomial is easily seen to be nonnegative everywhere. Further checking whether a polynomial is an SOS polynomial can be accomplished efficiently via semidefinite programming .
Turning our attention to the description of the convex hull of an algebraic variety, we will assume for the sake of simplicity that the convex hull is closed. Let be a polynomial ideal, and let be its real algebraic variety:
One can then show that the convex hull is given as:
A linear polynomial here is one that has a maximum degree of one, and the meaning of “modulo an ideal” is clear. As nonnegativity modulo an ideal may be intractable to check, we can consider a relaxation to a polynomial being SOS modulo an ideal, i.e., a polynomial that can be written as for in the ideal. Since it is tractable to check via semidefinite programmming whether bounded-degree polynomials are SOS, the -th theta body of an ideal is defined as follows in :
Here -sos refers to an SOS polynomial in which the components in the SOS decomposition have degree at most . The -th theta body is a convex relaxation of , and one can verify that