On the Quality of a Semidefinite Programming Bound for Sparse Principal Component Analysis

We examine the problem of approximating a positive, semidefinite matrix $\Sigma$ by a dyad $xx^T$, with a penalty on the cardinality of the vector $x$. This problem arises in sparse principal component analysis, where a decomposition of $\Sigma$ involving sparse factors is sought. We express this hard, combinatorial problem as a maximum eigenvalue problem, in which we seek to maximize, over a box, the largest eigenvalue of a symmetric matrix that is linear in the variables. This representation allows to use the techniques of robust optimization, to derive a bound based on semidefinite programming. The quality of the bound is investigated using a technique inspired by Nemirovski and Ben-Tal (2002).Comment: 13 pages, 3 figures This new version corresponds to an extensive revision of the earlier versio

Acknowledgement Research supported in part by NSF grant DMS-0625371. A Convex Upper Bound on the Log-Partition Function for Binary Graphical Models ∗

All rights reserved

A Semidefinite Relaxation for Air Traffic Flow Scheduling

We first formulate the problem of optimally scheduling air traffic low with sector capacity constraints as a mixed integer linear program. We then use semidefinite relaxation techniques to form a convex relaxation of that problem. Finally, we present a randomization algorithm to further improve the quality of the solution. Because of the specific structure of the air traffic flow problem, the relaxation has a single semidefinite constraint of size dn where d is the maximum delay and n the number of flights.Comment: Submitted to RIVF 200

Static Arbitrage Bounds on Basket Option Prices

We consider the problem of computing upper and lower bounds on the price of a European basket call option, given prices on other similar baskets. Although this problem is very hard to solve exactly in the general case, we show that in some instances the upper and lower bounds can be computed via simple closed-form expressions, or linear programs. We also introduce an efficient linear programming relaxation of the general problem based on an integral transform interpretation of the call price function. We show that this relaxation is tight in some of the special cases examined before.Comment: To Appear in Mathematical Programming, Series

Sparse Covariance Selection via Robust Maximum Likelihood Estimation

We address a problem of covariance selection, where we seek a trade-off between a high likelihood against the number of non-zero elements in the inverse covariance matrix. We solve a maximum likelihood problem with a penalty term given by the sum of absolute values of the elements of the inverse covariance matrix, and allow for imposing bounds on the condition number of the solution. The problem is directly amenable to now standard interior-point algorithms for convex optimization, but remains challenging due to its size. We first give some results on the theoretical computational complexity of the problem, by showing that a recent methodology for non-smooth convex optimization due to Nesterov can be applied to this problem, to greatly improve on the complexity estimate given by interior-point algorithms. We then examine two practical algorithms aimed at solving large-scale, noisy (hence dense) instances: one is based on a block-coordinate descent approach, where columns and rows are updated sequentially, another applies a dual version of Nesterov's method.Comment: Submitted to NIPS 200

Safe Feature Elimination for the LASSO and Sparse Supervised Learning Problems

We describe a fast method to eliminate features (variables) in l1 -penalized least-square regression (or LASSO) problems. The elimination of features leads to a potentially substantial reduction in running time, specially for large values of the penalty parameter. Our method is not heuristic: it only eliminates features that are guaranteed to be absent after solving the LASSO problem. The feature elimination step is easy to parallelize and can test each feature for elimination independently. Moreover, the computational effort of our method is negligible compared to that of solving the LASSO problem - roughly it is the same as single gradient step. Our method extends the scope of existing LASSO algorithms to treat larger data sets, previously out of their reach. We show how our method can be extended to general l1 -penalized convex problems and present preliminary results for the Sparse Support Vector Machine and Logistic Regression problems.Comment: Submitted to JMLR in April 201

Optimal Solutions for Sparse Principal Component Analysis

Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all target numbers of non zero coefficients, with total complexity O(n^3), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n^3) per pattern. We discuss applications in subset selection and sparse recovery and show on artificial examples and biological data that our algorithm does provide globally optimal solutions in many cases.Comment: Revised journal version. More efficient optimality conditions and new examples in subset selection and sparse recovery. Original version is in ICML proceeding

Model Selection Through Sparse Maximum Likelihood Estimation

We consider the problem of estimating the parameters of a Gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse. Our approach is to solve a maximum likelihood problem with an added l_1-norm penalty term. The problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes. We present two new algorithms for solving problems with at least a thousand nodes in the Gaussian case. Our first algorithm uses block coordinate descent, and can be interpreted as recursive l_1-norm penalized regression. Our second algorithm, based on Nesterov's first order method, yields a complexity estimate with a better dependence on problem size than existing interior point methods. Using a log determinant relaxation of the log partition function (Wainwright & Jordan (2006)), we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case. We test our algorithms on synthetic data, as well as on gene expression and senate voting records data

Robust sketching for multiple square-root LASSO problems

Many learning tasks, such as cross-validation, parameter search, or leave-one-out analysis, involve multiple instances of similar problems, each instance sharing a large part of learning data with the others. We introduce a robust framework for solving multiple square-root LASSO problems, based on a sketch of the learning data that uses low-rank approximations. Our approach allows a dramatic reduction in computational effort, in effect reducing the number of observations from $m$ (the number of observations to start with) to $k$ (the number of singular values retained in the low-rank model), while not sacrificing---sometimes even improving---the statistical performance. Theoretical analysis, as well as numerical experiments on both synthetic and real data, illustrate the efficiency of the method in large scale applications

First-order methods for sparse covariance selection

Given a sample covariance matrix, we solve a maximum likelihood problem penalized by the number of nonzero coefficients in the inverse covariance matrix. Our objective is to find a sparse representation of the sample data and to highlight conditional independence relationships between the sample variables. We first formulate a convex relaxation of this combinatorial problem, we then detail two efficient first-order algorithms with low memory requirements to solve large-scale, dense problem instances