Combinatorics and Optimizationhttp://hdl.handle.net/10012/99282019-07-17T19:27:24Z2019-07-17T19:27:24ZApplications of Stochastic Gradient Descent to Nonnegative Matrix FactorizationSlavin, Matthewhttp://hdl.handle.net/10012/147982019-07-16T02:31:03Z2019-07-15T00:00:00ZApplications of Stochastic Gradient Descent to Nonnegative Matrix Factorization
Slavin, Matthew
We consider the application of stochastic gradient descent (SGD) to the nonnegative matrix factorization (NMF) problem and the unconstrained low-rank matrix factorization problem. While the literature on the SGD algorithm is rich, the application of this specific algorithm to the field of matrix factorization problems is an unexplored area. We develop a series of results for the unconstrained problem, beginning with an analysis of standard gradient descent with a known zero-loss solution, and culminating with results for SGD in the general case where no zero-loss solution is assumed. We show that, with initialization close to a minimizer, there exist linear rate convergence guarantees.
We explore these results further with numerical experiments, and examine how the matrix factorization solutions found by SGD can be used as machine learning classifiers in two specific applications. In the first application, handwritten digit recognition, we show that our approach produces classification performance competitive with existing matrix factorization algorithms. In the second application, document topic classification, we examine how well SGD can recover an unknown words-to-topics matrix when the topics-to-document matrix is generated using the Latent Dirichlet Allocation model. This approach allows us to simulate two regimes for SGD: a fixed-sample regime where a large set of data is iterated over to train the model, and a generated-sample regime where a new data point is generated at each training iteration. In both regimes, we show that SGD can be an effective tool for recovering the hidden words-to-topic matrix. We conclude with some suggestions for further expansion of this work.
2019-07-15T00:00:00ZAlgebraic and combinatorial aspects of incidence groups and linear system non-local games arising from graphsPaddock, Connorhttp://hdl.handle.net/10012/147442019-06-07T02:30:28Z2019-06-06T00:00:00ZAlgebraic and combinatorial aspects of incidence groups and linear system non-local games arising from graphs
Paddock, Connor
To every linear binary-constraint system (LinBCS) non-local game, there is an associated algebraic object called the solution group. Cleve, Liu, and Slofstra showed that a LinBCS game has a perfect quantum strategy if and only if an element, denoted by $J$, is non-trivial in this group. In this work, we restrict to the set of graph-LinBCS games, which arise from $\mathbb{Z}_2$-linear systems $Ax=b$, where $A$ is the incidence matrix of a connected graph, and $b$ is a (non-proper) vertex $2$-colouring. In this context, Arkhipov's theorem states that the corresponding graph-LinBCS game has a perfect quantum strategy, and no perfect classical strategy, if and only if the graph is non-planar and the $2$-colouring $b$ has odd parity. In addition to efficient methods for detecting quantum and classical strategies for these games, we show that computing the classical value, a problem that is NP-hard for general LinBCS games can be done efficiently. In this work, we describe a graph-LinBCS game by a $2$-coloured graph and call the corresponding solution group a graph incidence group. As a consequence of the Robertson-Seymour theorem, we show that every quotient-closed property of a graph incidence group can be expressed by a finite set of forbidden graph minors. Using this result, we recover one direction of Arkhipov's theorem and derive the forbidden graph minors for the graph incidence group properties: finiteness, and abelianness. Lastly, using the representation theory of the graph incidence group, we discuss how our graph minor criteria can be used to deduce information about the perfect strategies for graph-LinBCS games.
2019-06-06T00:00:00ZThe Matching Augmentation Problem: A 7/4-Approximation AlgorithmDippel, Jackhttp://hdl.handle.net/10012/147002019-05-24T02:30:45Z2019-05-23T00:00:00ZThe Matching Augmentation Problem: A 7/4-Approximation Algorithm
Dippel, Jack
We present a 7/4 approximation algorithm for the matching augmentation problem (MAP): given a multi-graph with edges of cost either zero or one such that the edges of cost zero form a matching, find a 2-edge connected spanning subgraph (2-ECSS) of minimum cost. We first present a series of approximation guarantee preserving reductions, each of which can be performed in polytime. Performing these reductions gives us a restricted collection of MAP instances.
We present a 7/4 approximation algorithm for this restricted set of MAP instances. The algorithm starts with a subgraph which is a min-cost 2-edge cover, contracts its blocks, adds paths to the subgraph to cover all its bridges, and finally adds cycles to the subgraph to connect all its components. We contract any blocks created throughout. The algorithm ends when the subgraph is a single vertex, and we output all the edges we’ve contracted which form a 2ECSS.
2019-05-23T00:00:00ZNoisy Embezzlement of Entanglement and Applications to Entanglement DilutionLasecki, Dariuszhttp://hdl.handle.net/10012/146822019-05-24T02:32:00Z2019-05-23T00:00:00ZNoisy Embezzlement of Entanglement and Applications to Entanglement Dilution
Lasecki, Dariusz
In this thesis we present the concept of embezzlement of entanglement, its properties, efficiency, possible generalizations and propose the linear programming characterization of this phenomenon. Then, we focus on the noisy setting of embezzlement of entanglement. We provide the detailed proof of the quantum correlated sampling lemma which can be considered a protocol for noisy embezzlement of entanglement. Next, we propose a classical synchronization scheme for two spatially separated parties which do not communicate and use shared randomness to synchronize their descriptions of a quantum state. The result, together with the canonical embezzlement of entanglement, improves the quantum correlated sampling lemma for small quantum states in terms of the probability of success and distance between desired and final states. Then, we discuss the role of entanglement spread in dilution of entanglement. We propose an explicit protocol for the task of dilution of entanglement without communication. The protocol uses EPR pairs and an embezzling state of a relatively small size for the task of diluting entangled quantum states up to small infidelity. We modify the protocol to work in a noisy setting where the classical synchronization scheme finds its application.
2019-05-23T00:00:00Z