We describe an algorithm for nonlinear dimensionality reduction based on semidefinite programming and kernel matrix factorization. The algorithm learns a kernel matrix for high dimensional data that lies on or near a low dimensional manifold. In earlier work, the kernel matrix was learned by maximizing the variance in feature space while preserving the distances and angles between nearest neighbors. In this paper, adapting recent ideas from semi-supervised learning on graphs, we show that the full kernel matrix can be very well approximated by a product of smaller matrices. Representing the kernel matrix in this way, we can reformulate the semidefinite program in terms of a much smaller submatrix of inner products between randomly chosen landmarks. The new framework leads to order-of-magnitude reductions in computation time and makes it possible to study much larger problems in manifold learning. 1

Maximum variance unfolding (MVU) is an effective heuristic for dimensionality reduction. It produces a low-dimensional representation of the data by maximizing the variance of their embeddings while preserving the local distances of the original data. We show that MVU also optimizes a statistical dependence measure which aims to retain the identity of individual observations under the distancepreserving constraints. This general view allows us to design “colored ” variants of MVU, which produce low-dimensional representations for a given task, e.g. subject to class labels or other side information. 1

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It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained ℓ1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms ℓ1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted ℓ1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the ℓ1 norm of the coefficient sequence as is common, but by reweighting the ℓ1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as compressed sensing.

We present a unified duality view of several recently emerged spectral methods for nonlinear dimensionality reduction, including Isomap, locally linear embedding, Laplacian eigenmaps, and maximum variance unfolding. We discuss the duality theory for the maximum variance unfolding problem, and show that other methods are directly related to either its primal formulation or its dual formulation, or can be interpreted from the optimality conditions. This duality framework reveals close connections between these seemingly quite different algorithms. In particular, it resolves the myth about these methods in using either the top eigenvectors of a dense matrix, or the bottom eigenvectors of a sparse matrix — these two eigenspaces are exactly aligned at primal-dual optimality. 1.

We derive a stochastic gradient algorithm for semidefinite optimization using randomization techniques. The algorithm uses subsampling to reduce the computational cost of each iteration and the subsampling ratio explicitly controls the algorithm’s granularity, i.e. the tradeoff between cost per iteration and total number of iterations. Furthermore, the total computational cost is directly proportional to the complexity (i.e. rank) of the solution. We study numerical performance on some large-scale problems arising in statistical learning.

Abstract. The effective resistance between two nodes of a weighted graph is the electrical resistance seen between the nodes of a resistor network with branch conductances given by the edge weights. The effective resistance comes up in many applications and fields in addition to electrical network analysis, including, for example, Markov chains and continuous-time averaging networks. In this paper we study the problem of allocating edge weights on a given graph in order to minimize the total effective resistance, i.e., the sum of the resistances between all pairs of nodes. We show that this is a convex optimization problem, and can be solved efficiently either numerically, or, in some cases, analytically. We show that optimal allocation of the edge weights can reduce the total effective resistance of the graph (compared to uniform weights) by a factor that grows unboundedly with the size of the graph. We show that among all graphs with n nodes, the path has the largest value of optimal total effective resistance, and the complete graph the least. 1. Introduction. Let N be a network with n nodes and m edges, i.e., an undirected graph (V, E) with |V | = n, |E | = m, and nonnegative weights on the edges. We call the weight on edge l its conductance, and denote it by gl. The effective resistance between a pair of nodes i and j, denoted Rij, is the electrical resistance measured across nodes i and j, when the network represents an electrical circuit with each edge (or branch, in the terminology of electrical circuits) a resistor with (electrical) conductance gl. In other

Abstract. We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this problem is convex, i.e., it involves minimizing a convex function (or maximizing a concave function) over a convex set. This allows us to give simple necessary and sufficient optimality conditions, derive interesting dual problems, find analytical solutions in some cases, and efficiently compute numerical solutions in all cases. In this overview we briefly describe some more specific cases of this general problem, which have been addressed in a series of recent papers. • Fastest mixing Markov chain. Find edge transition probabilities that give the fastest mixing (symmetric, discrete-time) Markov chain on the graph. • Fastest mixing Markov process. Find the edge transition rates that give the fastest mixing (symmetric, continuous-time) Markov process on the graph. • Absolute algebraic connectivity. Find edge weights that maximize the algebraic

This paper identifies a broad class of nonlinear dimensionality reduced (NLDR) problems where the exact local isometry between an extrinsically curved data manifold M and a low-dimensional parameterization space can be recovered from a finite set of high-dimensional point sampels. The method, Geodesic Nullsapce Analysis (GNA), rests on two results: First, the exact isometric parameterization of a local point clique on M haas an algebraic reduction to arc-length integrations when the ambient-space embedding of M is locally a product of planar quadrics. Second, the locally isometric global parameterization lies in the left invariant subspace of a linearizing operator that averages the nullspace projectors of the local parameterizations. We show how to use the GNA operator for denosing, dimensionality reduction, and resynthesis of both the original data and of new samples, making such "submanifold" methods an attractive alternative to subspace methods in data analysis.