Descent methods for Nonnegative Matrix Factorization

In this paper, we present several descent methods that can be applied to nonnegative matrix factorization and we analyze a recently developped fast block coordinate method called Rank-one Residue Iteration (RRI). We also give a comparison of these different methods and show that the new block coordinate method has better properties in terms of approximation error and complexity. By interpreting this method as a rank-one approximation of the residue matrix, we prove that it \emph{converges} and also extend it to the nonnegative tensor factorization and introduce some variants of the method by imposing some additional controllable constraints such as: sparsity, discreteness and smoothness.Comment: 47 pages. New convergence proof using damped version of RRI. To appear in Numerical Linear Algebra in Signals, Systems and Control. Accepted. Illustrating Matlab code is included in the source bundl

Algorithms, applications and systems towards interpretable pattern mining from multi-aspect data

How do humans move around in the urban space and how do they differ when the city undergoes terrorist attacks? How do users behave in Massive Open Online courses~(MOOCs) and how do they differ if some of them achieve certificates while some of them not? What areas in the court elite players, such as Stephen Curry, LeBron James, like to make their shots in the course of the game? How can we uncover the hidden habits that govern our online purchases? Are there unspoken agendas in how different states pass legislation of certain kinds? At the heart of these seemingly unconnected puzzles is this same mystery of multi-aspect mining, i.g., how can we mine and interpret the hidden pattern from a dataset that simultaneously reveals the associations, or changes of the associations, among various aspects of the data (e.g., a shot could be described with three aspects, player, time of the game, and area in the court)? Solving this problem could open gates to a deep understanding of underlying mechanisms for many real-world phenomena. While much of the research in multi-aspect mining contribute broad scope of innovations in the mining part, interpretation of patterns from the perspective of users (or domain experts) is often overlooked. Questions like what do they require for patterns, how good are the patterns, or how to read them, have barely been addressed. Without efficient and effective ways of involving users in the process of multi-aspect mining, the results are likely to lead to something difficult for them to comprehend. This dissertation proposes the M^3 framework, which consists of multiplex pattern discovery, multifaceted pattern evaluation, and multipurpose pattern presentation, to tackle the challenges of multi-aspect pattern discovery. Based on this framework, we develop algorithms, applications, and analytic systems to enable interpretable pattern discovery from multi-aspect data. Following the concept of meaningful multiplex pattern discovery, we propose PairFac to close the gap between human information needs and naive mining optimization. We demonstrate its effectiveness in the context of impact discovery in the aftermath of urban disasters. We develop iDisc to target the crossing of multiplex pattern discovery with multifaceted pattern evaluation. iDisc meets the specific information need in understanding multi-level, contrastive behavior patterns. As an example, we use iDisc to predict student performance outcomes in Massive Open Online Courses given users' latent behaviors. FacIt is an interactive visual analytic system that sits at the intersection of all three components and enables for interpretable, fine-tunable, and scrutinizable pattern discovery from multi-aspect data. We demonstrate each work's significance and implications in its respective problem context. As a whole, this series of studies is an effort to instantiate the M^3 framework and push the field of multi-aspect mining towards a more human-centric process in real-world applications

LASS: a simple assignment model with Laplacian smoothing

We consider the problem of learning soft assignments of $N$ items to $K$ categories given two sources of information: an item-category similarity matrix, which encourages items to be assigned to categories they are similar to (and to not be assigned to categories they are dissimilar to), and an item-item similarity matrix, which encourages similar items to have similar assignments. We propose a simple quadratic programming model that captures this intuition. We give necessary conditions for its solution to be unique, define an out-of-sample mapping, and derive a simple, effective training algorithm based on the alternating direction method of multipliers. The model predicts reasonable assignments from even a few similarity values, and can be seen as a generalization of semisupervised learning. It is particularly useful when items naturally belong to multiple categories, as for example when annotating documents with keywords or pictures with tags, with partially tagged items, or when the categories have complex interrelations (e.g. hierarchical) that are unknown.Comment: 20 pages, 4 figures. A shorter version appears in AAAI 201

Formation Shape Control Based on Distance Measurements Using Lie Bracket Approximations

We study the problem of distance-based formation control in autonomous multi-agent systems in which only distance measurements are available. This means that the target formations as well as the sensed variables are both determined by distances. We propose a fully distributed distance-only control law, which requires neither a time synchronization of the agents nor storage of measured data. The approach is applicable to point agents in the Euclidean space of arbitrary dimension. Under the assumption of infinitesimal rigidity of the target formations, we show that the proposed control law induces local uniform asymptotic stability. Our approach involves sinusoidal perturbations in order to extract information about the negative gradient direction of each agent's local potential function. An averaging analysis reveals that the gradient information originates from an approximation of Lie brackets of certain vector fields. The method is based on a recently introduced approach to the problem of extremum seeking control. We discuss the relation in the paper

Quadratic-exponential coherent feedback control of linear quantum stochastic systems

This paper considers a risk-sensitive optimal control problem for a field-mediated interconnection of a quantum plant with a coherent (measurement-free) quantum controller. The plant and the controller are multimode open quantum harmonic oscillators governed by linear quantum stochastic differential equations, which are coupled to each other and driven by multichannel quantum Wiener processes modelling the external bosonic fields. The control objective is to internally stabilize the closed-loop system and minimize the infinite-horizon asymptotic growth rate of a quadratic-exponential functional which penalizes the plant variables and the controller output. We obtain first-order necessary conditions of optimality for this problem by computing the partial Frechet derivatives of the cost functional with respect to the energy and coupling matrices of the controller in frequency domain and state space. An infinitesimal equivalence between the risk-sensitive and weighted coherent quantum LQG control problems is also established. In addition to variational methods, we employ spectral factorizations and infinite cascades of auxiliary classical systems. Their truncations are applicable to numerical optimization algorithms (such as the gradient descent) for coherent quantum risk-sensitive feedback synthesis.Comment: 29 pages, 3 figure

New optimization methods in predictive control

This thesis is mainly concerned with the efficient solution of a linear discrete-time finite horizon optimal control problem (FHOCP) with quadratic cost and linear constraints on the states and inputs. In predictive control, such a FHOCP needs to be solved online at each sampling instant. In order to solve such a FHOCP, it is necessary to solve a quadratic programming (QP) problem. Interior point methods (IPMs) have proven to be an efficient way of solving quadratic programming problems. A linear system of equations needs to be solved in each iteration of an IPM. The ill-conditioning of this linear system in the later iterations of the IPM prevents the use of an iterative method in solving the linear system due to a very slow rate of convergence; in some cases the solution never reaches the desired accuracy. A new well-conditioned IPM, which increases the rate of convergence of the iterative method is proposed. The computational advantage is obtained by the use of an inexact Newton method along with the use of novel preconditioners. A new warm-start strategy is also presented to solve a QP with an interior-point method whose data is slightly perturbed from the previous QP. The effectiveness of this warm-start strategy is demonstrated on a number of available online benchmark problems. Numerical results indicate that the proposed technique depends upon the size of perturbation and it leads to a reduction of 30-74% in floating point operations compared to a cold-start interior point method. Following the main theme of this thesis, which is to improve the computational efficiency of an algorithm, an efficient algorithm for solving the coupled Sylvester equation that arises in converting a system of linear differential-algebraic equations (DAEs) to ordinary differential equations is also presented. A significant computational advantage is obtained by exploiting the structure of the involved matrices. The proposed algorithm removes the need to solve a standard Sylvester equation or to invert a matrix. The improved performance of this new method over existing techniques is demonstrated by comparing the number of floating-point operations and via numerical examples