Consider the problem of estimating an covariance matrix (or its inverse ) of a
-variate probability distribution fromindependently and identically distributed samples 24, 25, 10], the matrix is often sparse
, meaning that its matrix elements are mostly zero. For Gaussian distributions, the statistical interpretation of sparsity inis that most of the variables are pairwise conditionally independent [27, 45, 14, 5].
Imposing sparsity upon can regularize the associated estimation problem and greatly reduce the number of samples required. This is particularly important in high-dimensional settings where is large, often significantly larger than the number of samples . One popular approach regularizes the associated maximum likelihood estimation (MLE) problem by a sparsity-promoting term, as in
Here, is the sample covariance matrix with sample mean , and is the resulting estimator for . This approach, commonly known as the graphical lasso , is known to enjoy a number of statistical guarantees [37, 34], some of which are direct extensions of earlier work on the classical lasso [31, 29, 42, 22]. A variation on this theme is to only impose the penalty on the off-diagonal elements of , or to place different weights on the elements of the matrix , as in the classical weighted lasso.
While the -regularized problem (1) is technically convex, it is commonly considered intractable for large-scale datasets. The decision variable is an matrix, so simply fitting all variables into memory is already a significant issue. General-purpose algorithms have either prohibitively high complexity or slow convergence. In practice, (1) is solved using problem-specific algorithms. The state-of-the-art include GLASSO , QUIC , and its “big-data” extension BIG-QUIC . These algorithms use between and time and between and memory per iteration, but the number of iterations needed to converge to an accurate solution can be very large.
1.1 Graphical lasso, soft-thresholding, and MDMC
The high practical cost of graphical lasso has inspired a number of heuristics, which enjoy less guarantees but are significantly cheaper to use. Indeed, heuristics are often the only viable option onceexceeds the order of a few tens of thousands.
One simple idea is to threshold the sample covariance matrix : to examine all of its elements and keep only the ones whose magnitudes exceed some threshold. Thresholding can be fast—even for very large-scale datasets—because it is embarassingly parallel; its quadratic total work can be spread over thousands or millions of parallel processors, in a GPU or distributed on cloud servers. When the number of samples is small, i.e. , thresholding can also be performed using memory, by working directly with the centered matrix-of-samples satisfying .
In a recent line of work [26, 39, 12, 13], the simple heuristic of thresholding was shown to enjoy some surprising guarantees. In particular, [39, 12] proved that when the lasso weight is imposed over only the off-diagonal elements of that—under some assumptions—the sparsity pattern of the associated graphical lasso estimator can be recovered by performing a soft-thresholding operation on , as in
The associated graph (also denoted as when there is no ambiguity) is obtained by viewing each nonzero element in as an edge between the -th and -th vertex in an undirected graph on nodes. Moreover, they showed that the estimator can be recovered by solving a version of (1) in which the sparsity pattern is explicitly imposed, as in
Recovering the exact value of (and not just its sparsity pattern) is important because it provides a shrinkage MLE when the true MLE is ill-defined; for Gaussian fields, its nonzero values encode the partial correlations between variables. Problem (4) is named the maximum determinant matrix completion (MDMC) in the literature, for reasons explained below. The problem has a recursive closed-form solution whenever the graph of is acyclic (i.e. a tree or forest) , or more generally, if it is chordal . It is worth emphasizing that the closed-form solution is extremely fast to evaluate: a chordal example in  with 13,659 variables took just seconds to solve on a laptop computer.
The assumptions needed for graphical lasso to be equivalent to thresolding are hard to check but relatively mild. Indeed,  proves that they are automatically satisfied whenever is sufficiently large relative to the sample covariance matrix. Their numerical study found “sufficiently large” to be a fairly loose criterion in practice, particularly in view of the fact that large values of are needed to induce a sufficiently sparse estimate of , e.g. with nonzero elements.
However, the requirement for to be chordal is very strong. Aside from trivial chordal graphs like trees and cliques, thresholding will produce a chordal graph with probability zero. When is nonchordal, no closed-form solution exists, and one must resort to an iterative algorithm. The state-of-the-art for nonchordal MDMC is to embed the nonchordal graph within a chordal graph, and to solve the resulting problem as a semidefinite program using an interior-point method [8, 2].
1.2 Main results
The purpose of this paper is two-fold. First, we derive an extension of the guarantees derived by [26, 39, 12, 13] for a slightly more general version of the problem that we call restricted graphical lasso (RGL):
In other words, RGL is (1) penalized by a weighted lasso penalty on the off-diagonals, and with an a priori sparsity pattern imposed as an additional constraint. We use the sparsity pattern to incorporate prior information on the structure of the graphical model. For example, if the sample covariance is collected over a graph, such as a communication system or a social network, then far-away variables can be assumed as pairwise conditionally independent [33, 19, 7]. Including these neighborhood relationships into can regularize the statistical problem, as well as reduce the numerical cost for a solution.
In Section 2, we describe a procedure to transform RGL (5) into MDMC (4), in the same style as prior results by [12, 13] for graphical lasso. More specifically, we soft-threshold the sample covariance and then project this matrix onto the sparsity pattern . We give conditions for the resulting sparsity pattern to be equivalent to the one obtained by solving (5). Furthermore, we prove that the resulting estimator can be recovered by solving the same MDMC problem (4) with appropriately modified.
The second purpose is to describe an efficient algorithm to solve MCDC when the graph is nonchordal, based on the chordal embedding approach of [8, 2, 4]. We embed within a chordal , to result in a convex optimization problem over , the space of real symmetric matrices with sparsity pattern . This way, the constraint is implicitly imposed, meaning that we simply ignore the nonzero elements not in . Next, we solve an optimization problem on using a custom Newton-CG method111The MATLAB source code for our solver can be found at http://alum.mit.edu/www/ryz. The main idea is to use an inner conjugate gradients (CG) loop to solve the Newton subproblem of an outer Newton’s method. The actual algorithm has a number of features designed to exploit problem structure, including the sparse chordal property of , duality, and the ability for CG and Newton to converge superlinearly; these are outlined in Section 3.
Most importantly, the algorithm is highly efficient in practice. In Section 4, we present computation results on a suite of test cases. Both synthetic and real-life graphs are considered. Using our approach, we solve sparse inverse covariance estimation problems containing as many as 200,000 variables, in less than an hour on a laptop computer.
1.3 Related Work
Graphical lasso with prior information. A number of approaches are available in the literature to introduce prior information to graphical lasso. The weighted version of graphical lasso mentioned before is an example, though RGL will generally be more efficient to solve due to a reduction in the number of variables.  introduced a class of graphical lasso in which the true graphical model is assumed to have Laplacian structure. This structure commonly appears in signal and image processing . For the a priori graph-based correlation structure described above,  introduced a pathway graphical lasso method similar to RGL.
Algorithms for graphical lasso. Algorithms for graphical lasso are usually based on some mixture of Newton , proximal Newton [21, 20], iterative thresholding , and (block) coordinate descent [14, 40]. All of these suffer fundamentally from the need to keep track and act on all elements in the matrix decision variable. Even if the final solution matrix were sparse with nonzeros, it is still possible for the algorithm to traverse through a “dense region” in which the iterate must be fully dense. Thresholding heuristics have been proposed to address issue, but these may adversely affect the outer algorithm and prevent convergence. It is generally impossible to guarantee a figure lower than time per-iteration, even if the solution contains only nonzeros. Most of the algorithms mentioned above actually have worst-case per-iteration costs of .
Graphical lasso via thresholding. The elementary estimator for graphical models (EE-GM)  is another thresholding-based low-complexity method that is able to recover the actual graphical lasso estimator. Both EE-GM and our algorithm have a similar level of performance in practice, because both algorithm are bottlenecked by the initial thresholding step, which is a quadratic time operation.
Algorithms for MDMC. Our algorithm is inspired by a line of results [8, 2, 4, 23] for minimizing the log-det penalty on chordal sparsity patterns, culminating in the CVXOPT package . These are Newton algorithms that solve the Newton subproblem by explicitly forming and factoring the fully-dense Newton matrix. When , these algorithms cost time and memory per iteration, where is the number of edges added to to yield the chordal . In practice, is usually a factor of 0.1 to 20 times , so these algorithms are cubic time and memory. Our algorithm solves the Newton subproblem iteratively using CG. We prove that CG requires just time to compute the Newton direction to machine precision (see Section 3.4
). In practice, CG converges much faster than its worst-case bound, because it is able to exploit eigenvalue clustering to achieve superlinear convergence.
Let be the set of
real vectors, andbe the set of real symmetric matrices. (We denote using lower-case, using upper-case, and index the -th element of as .) We endow with the usual matrix inner product and Euclidean (i.e. Frobenius) norm . Let and be the associated set of positive semidefinite and positive definite matrices. We will frequently write to mean and write to mean . Given a sparsity pattern , we define as the set of real symmetric matrices with this sparsity pattern.
2 Restricted graphical lasso, soft-thresholding, and MDMC
Let denote the projection operator from onto , i.e. by setting all if . Let be the sample covariance matrix individually soft-thresholded by , as in
In this section, we state the conditions for —the projection of the soft-thresholded matrix in (7) onto —to have the same sparsity pattern as the RGL estimator in (5). Furthermore, the estimator can be explicitly recovered by solving the MDMC problem (4) while replacing and . For brevity, all proofs and remarks are omitted; these can be found in the appendix.
Before we state the exact conditions, we begin by adopting the some definitions and notations from the literature.
 Given a matrix , define as its sparsity pattern. Then is called inverse-consistent if there exists a matrix such that
The matrix is called an inverse-consistent complement of and is denoted by . Furthermore, is called sign-consistent if for every , the -th elements of and have opposite signs.
Moreover, we take the usual matrix max-norm to exclude the diagonal, as in and adopt the function defined with respect to the sparsity pattern and scalar
We are now ready to state the conditions for soft-thresholding to be equivalent to RGL.
See Appendix A. ∎
Theorem 2 leads to the following corollary, which asserts that the optimal solution of RGL can be obtained by maximum determinant matrix completion: computing the matrix with the largest determinant that “fills-in” the zero elements of .
3 Proposed Algorithm
This section describes an efficient algorithm to solve MDMC (4) in which the sparsity pattern is nonchordal. If we assume that the input matrix is sparse, and that sparse Cholesky factorization is able to solve in time, then our algorithm is guaranteed to compute an -accurate solution in time and memory.
The algorithm is fundamentally a Newton-CG method, i.e. Newton’s method in which the Newton search directions are computed using conjugate gradients (CG). It is developed from four key insights:
1. Chordal embedding is easy via sparse matrix heuristics. State-of-the-art algorithms for (4) begin by computing a chordal embedding for . The optimal chordal embedding with the fewest number of nonzeros is NP-hard to compute, but a good-enough embedding with nonzeros is sufficient for our purposes. Computing a good with is exactly the same problem as finding a sparse Cholesky factorization with fill-in. Using heuristics developed for numerical linear algebra, we are able to find sparse chordal embeddings for graphs containing millions of edges and hundreds of thousands of nodes in seconds.
2. Optimize directly on the sparse matrix cone. Using log-det barriers for sparse matrix cones [8, 2, 4, 41], we can optimize directly in the space , while ignoring all matrix elements outside of . If , then only decision variables must be explicitly optimized. Moreover, each function evaluation, gradient evaluation, and matrix-vector product with the Hessian can be performed in time, using the numerical recipes in .
3. The dual is easier to solve than the primal. The primal problem starts with a feasible point and seeks to achieve first-order optimality. The dual problem starts with an infeasible optimal point satisfying first-order optimality, and seeks to make it feasible. Feasibility is easier to achieve than optimality, so the dual problem is easier to solve than the primal.
4. Conjugate gradients (CG) converges in iterations. Under the same conditions that allow Theorem 2 to work, our main result (Theorem 6) bounds the condition number of the Newton subproblem to be , independent of the problem dimension and the current accuracy . It is therefore cheaper to solve this subproblem using CG to machine precision in time than it is to solve for it directly in time using Cholesky factorization [8, 2, 4]. Moreover, CG is an optimal Krylov subspace method, and as such, it is often able to exploit clustering in the eigenvalues to converge superlinearly. Finally, computing the Newton direction to high accuracy further allows the outer Newton method to also converge quadratically.
The remainder of this section describes each consideration in further detail. We state the algorithm explicitly in Section 3.5.
3.1 Efficient chordal embedding
in which is a chordal embedding for : a sparsity pattern whose graph contains no induced cycles greater than three. This can be implemented using standard algorithms for large-and-sparse linear equations, due to the following result.
Let be a positive definite matrix with sparsity pattern . Compute its unique lower-triangular Cholesky factor satisfying . Ignoring perfect numerical cancellation, the sparsity pattern of is a chordal embedding .
Note that can be determined directly from using a symbolic Cholesky algorithm, which simulates the steps of Gaussian elimination using Boolean logic. Moreover, we can substantially reduce the number of edges added to by reordering the columns and rows of using a fill-reducing ordering.
Let be a permutation matrix. For the same in Proposition 4, compute the unique Cholesky factor satisfying . Ignoring perfect numerical cancellation, the sparsity pattern of is a chordal embedding .
The problem of finding the best choice of is known as the fill-minimizing problem, and is NP-complete . However, good orderings are easily found using heuristics developed for numerical linear algebra, like minimum degree ordering  and nested dissection [16, 1]. In fact,  proved that nested dissection is suboptimal for bounded-degree graphs, and notes that “we do not know a class of graphs for which [nested dissection is suboptimal] by more than a constant factor.”
If admits sparse chordal embeddings, then a good-enough will usually be found using minimum degree or nested dissection. In MATLAB, the minimum degree ordering and symbolic factorization steps can be performed in two lines of code; see the snippet in Figure 1.
3.2 Logarithmic barriers for sparse matrix cones
Define the cone of sparse positive semidefinite matrices , and the cone of sparse matrices with positive semidefinite completions , as the following
Then (12) can be posed as the primal-dual pair:
The linear map converts a list of variables into the corresponding matrix in . The gradients of are simply the projections of their usual values onto , as in
Given any let be the unique matrix satisfying . Then we have
Assuming that is sparse and chordal, all six operations can be efficiently evaluated in time and memory, using the numerical recipes described in .
3.3 Solving the dual problem
Our algorithm actually solves the dual problem (14), which can be rewritten as an unconstrained optimization problem
After the solution is found, we can recover the optimal estimator for the primal problem via . The dual problem (14) is easier to solve than the primal (13) because the origin often lies very close to the solution . To see this, note that produces a candidate estimator that solves the chordal matrix completion problem
which is a relaxation of the nonchordal problem posed over . As observed by several previous authors , this relaxation is a high quality guess, and is often “almost feasible” for the original nonchordal problem posed over , as in . Some simple algebra shows that the gradient evaluated at the origin has Euclidean norm , so if holds true, then the origin is close to optimal. Starting from this point, we can expect Newton’s method to rapidly converge at a quadratic rate.
3.4 CG converges in iterations
The most computationally expensive part of Newton’s method is the solution of the Newton direction via the system of equations
The Hessian matrix is fully dense, but matrix-vector products are linear time using the algorithms in Section LABEL:subsec:Logarithmic-barriers-for. This insight motivates solving (16) using an iterative Krylov subspace method like conjugate gradients (CG), which is a matrix-free method that requires a single matrix-vector product with at each iteration . Starting from the origin , the method converges to an -accurate search direction satisfying
in at most
where is the condition number of the Hessian matrix [18, 38]. In many important convex optimization problems, the condition number grows like or as the outer Newton iterates approach an -neighborhood of the true solution. As a consequence, Newton-CG methods typically require or CG iterations.
It is therefore surprising that we are able to bound globally for the MDMC problem. Below, we state our main result, which says that the condition number depends polynomially on the problem data and the quality of the initial point, but is independent of the problem dimension and the accuracy of the current iterate .
At any satisfying and , the condition number of the Hessian matrix is bound
is the suboptimality of the initial point,
is the vectorized version of the problem data,
and are the initial primal-dual pair,
and are the solution primal-dual pair.
See Appendix B. ∎
Newton’s method is a descent method, so its -th iterate trivially satisfies . Technically, the condition can be guaranteed by enclosing Newton’s method within an outer auxillary path-following loop; see Section 4.3.5 of . In practice, naive Newton’s method will usually satisfy the condition on its own; see our numerical experiments in Section 4.
Multiplying this figure by the cost of each CG iteration proves the claimed time complexity in (6). In practice, CG typically converges much faster than this worst-case bound, due to its ability to exploit the clustering of eigenvalues in ; see [18, 38]. Moreover, accurate Newton directions are only needed to guarantee quadratic convergence close to the solution. During the initial Newton steps, we may loosen the error tolerance for CG for a significant speed-up. Inexact Newton steps can be used to obtain a speed-up of a factor of 2-3.
3.5 The full algorithm
To summarize, we begin by computing a chordal embedding for the sparsity pattern of , using the code snippet in Figure 1. We use the embedding to reformulate (4) as (12), and solve the unconstrained problem defined in (15), using Newton’s method
starting at the origin . The function value , gradient and Hessian matrix-vector products are all evaluated using the numerical recipes described by .
At each -th Newton step, we compute the Newton search direction using conjugate gradients. A loose tolerance is used when the Newton decrement is large, and a tight tolerance is used when the decrement is small, implying that the iterate is close to the true solution.
Once a Newton direction is computed with a sufficiently large Newton decrement , we use a backtracking line search to determine the step-size . In other words, we select the first instance of the sequence that satisfies the Armijo–Goldstein condition
in which and are line search parameters. Our implementation used and . We complete the step and repeat the process, until convergence.
We terminate the outer Newton’s method if the Newton decrement falls below a threshold. This implies either that the solution has been reached, or that CG is not converging to a good enough to make significant progress. The associated estimator for is recovered by evaluating .
4 Numerical Results
Finally, we benchmark our algorithm222The MATLAB source code for our solver can be found at http://alum.mit.edu/www/ryz against QUIC , commonly considered the fastest solver for graphical lasso or RGL333QUIC was taken from http://bigdata.ices.utexas.edu/software/1035/. (Another widely-used algorithm is GLASSO , but we found it to be significantly slower than QUIC.) We consider two case studies. The first case study numerically verifies the claimed complexity of our MDMC algorithm on problems with a nearly-banded structure. The second case study performs the full threshold-MDMC procedure for graphical lasso and RGL, on graphs collected from real-life applications.
All experiments are performed on a laptop computer with an Intel Core i7 quad-core 2.50 GHz CPU and 16GB RAM. The reported results are based on a serial implementation in MATLAB-R2017b. Both our Newton decrement threshold and QUIC’s convergence threshold are .
We implemented the soft-thresholding set (7) as a serial routine that uses memory by taking the matrix-of-samples satisfying as input. The routine implicitly partitions into submatrices of size , and iterates over the submatrices one at a time. For each submatrix, it explicitly forms the submatrix, thresholds it using dense linear algebra, and then stores the result as a sparse matrix.
4.1 Case Study 1: Banded Patterns
The first case study aims to verify the claimed complexity of our algorithm for MDMC. Here, we avoid the proposed thresholding step, and focus solely on the MDMC (4) problem. Each sparsity pattern is a corrupted banded matrices with bandwidth . The off-diagonal nonzero elements of
are selected from the uniform distribution inand then corrupted to zero with probability 0.3. The diagonal elements are fixed to . Our numerical experiments fix the bandwidth and vary the number of variables from 1,000 to 200,000. A time limit of 2 hours is set for both algorithms.
Figure 2a compares the running time of both algorithms. A log-log regression results in an empirical time complexity of for our algorithm, and for QUIC. The extra 0.1 in the exponent is most likely an artifact our MATLAB implementation. In either case, QUIC’s quadratic complexity limits it to . By contrast, our algorithm solves an instance with in less than 33 minutes. The resulting solutions are extremely accurate, with optimality and feasibility gaps of less than and , respectively.
4.2 Case Study 2: Real-Life Graphs
|#||file name||type||sec||gap||feas||sec||diff. gap||speed-up|
The second case study aims to benchmark the full thresholding-MDMC procedure for sparse inverse covariance estimation on real-life graphs. The actual graphs (i.e. the sparsity patterns) for are chosen from SuiteSparse Matrix Collection —a publicly available dataset for large-and-sparse matrices collected from real-world applications. Our chosen graphs vary in size from to , and are taken from applications in chemical processes, material science, graph problems, optimal control and model reduction, thermal processes and circuit simulations.
For each sparsity pattern , we design a corresponding as follows. For each , we select from the uniform distribution in , and then corrupt it to zero with probability . Then, we set each diagonal to . Using this , we generate samples i.i.d. as . This results in a sample covariance matrix .
We solve graphical lasso and RGL with the described above using our proposed soft-thresholding-MDMC algorithm and QUIC, in order to estimate . In the case of RGL, we assume that the graph is known a priori, while noting that 30% of the elements of have been corrupted to zero. Our goal here is to discover the location of these corrupted elements. In all of our simulations, the threshold is set so that the number of nonzero elements in the the estimator is roughly the same as the ground truth. We limit both algorithms to 3 hours of CPU time.
Figure 2b compares the CPU time of both two algorithms for this case study; the specific details are provided in Table 1. A log-log regression results in an empirical time complexity of and for graphical lasso and RGL using our algorithm, and and for the same using QUIC. The exponents of our algorithm are due to the initial soft-thresholding step, which is quadratic-time on a serial computer, but because the overall procedure is dominated by the solution of the MDMC. Both algorithms solve graphs with within the allotted time limit, though our algorithm is 11 times faster on average. Only our algorithm is able to solve the estimation problem with in a little more than an hour.
To check whether thresholding-MDMC really does solve graphical lasso and RGL, we substitute the two sets of estimators back into their original problems (1) and (5). The corresponding objective values have a relative difference , suggesting that both sets of estimators are about equally optimal. This observation verifies our claims in Theorem 2 and Corollary 3 that (1) and (5): thresholding-MDMC does indeed solve graphical lasso and RGL.
Graphical lasso is a widely-used approach for estimating a covariance matrix with a sparse inverse from limited samples. In this paper, we consider a slightly more general formulation called restricted graphical lasso (RGL), which additionally enforces a prior sparsity pattern to the estimation. We describe an efficient approach that substantially reduces the cost of solving RGL: 1) soft-thresholding the sample covariance matrix and projecting onto the prior pattern, to recover the estimator’s sparsity pattern; and 2) solving a maximum determinant matrix completion (MDMC) problem, to recover the estimator’s numerical values. The first step is quadratic time and memory but embarrassingly parallelizable. If the resulting sparsity pattern is sparse and chordal, then under mild technical assumptions, the second step can be performed using the Newton-CG algorithm described in this paper in linear time and memory. The algorithm is tested on both synthetic and real-life data, solving instances with as many as 200,000 variables to 7-9 digits of accuracy within an hour on a standard laptop computer.
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Appendix A Restricted graphical Lasso and MDMC
Our aim is to elucidate the connection between the RGL and MDMC problem under the assumption that the regularization coefficients are chosen to be large, i.e., when a sparse solution for the RGL is sought. Recall that RGL is formulated as follows
Now, consider the following modified soft-thresholded sample covariance matrix
Definition 8 ().
For a given symmetric matrix , let denote the minimal set such that . is called inverse-consistent if there exists a matrix with zero diagonal such that
The matrix is called an inverse-consistent complement of and is denoted by . Furthermore, is called sign-consistent if for every , the entries of and have opposite signs.
Definition 9 ().
Given a sparsity pattern and a scalar , define as the maximum of over all inverse-consistent positive-definite matrices with the diagonal entries all equal to 1 such that and .
Without loss of generality, we make the following mild assumption.
and for every .
We are now ready to restate Theorem 2.
First, note that the diagonal elements of are and its off-diagonal elements are between and . A sparse solution for RGL requires large regularization coefficients. This leads to numerous zero elements in and forces the magnitude of the nonzero elements to be small. This means that, in most instances, is positive definite or even diagonally dominant. Certifying Condition (ii) is hard in general. However,  shows that this condition is automatically implied by Condition (i) when induces an acyclic structure. More generally,  shows that is sign-consistent if is close to its first order Taylor expansion. This assumption holds in practice due to the fact that the magnitude of the off-diagonal elements of is small. Furthermore,  proves that this condition is necessary for the equivalence between the sparsity patterns of thresholding and GL when the regularization matrix is large enough. Finally,  shows that the left hand side of (9) is upper bounded by for some which only depends on . This implies that, when is small, or equivalently the regularization matrix is large, Condition (iii) is automatically satisfied.
In this section, we present the technical proofs of our theorems. To prove Theorem 2, we need a number of lemmas. First, consider the RGL problem, with . The first lemma offers optimality (KKT) conditions for the unique solution of this problem.
is the optimal solution of RGL problem with if and only if it satisfies the following conditions for every :
where denotes the entry of .
The proof is straightforward and omitted for brevity. ∎
Now, consider the following optimization:
Let denotes the optimal solution of (23). Furthermore, define as a diagonal matrix with for every . The following lemma relates to .
We have .
To prove this lemma, we define an intermediate optimization problem. Consider