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Concept

Matrix completion

Résumé

Matrix completion is the task of filling in the missing entries of a partially observed matrix, which is equivalent to performing data imputation in statistics. A wide range of datasets are naturally organized in matrix form. One example is the movie-ratings matrix, as appears in the Netflix problem: Given a ratings matrix in which each entry (i,j) represents the rating of movie j by customer i, if customer i has watched movie j and is otherwise missing, we would like to predict the remaining entries in order to make good recommendations to customers on what to watch next. Another example is the document-term matrix: The frequencies of words used in a collection of documents can be represented as a matrix, where each entry corresponds to the number of times the associated term appears in the indicated document. Without any restrictions on the number of degrees of freedom in the completed matrix this problem is underdet

Source officielle

https://en.wikipedia.org/wiki/Matrix_completion

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Séances de cours associées (5)

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Traditional approaches to analysing functional data typically follow a two-step procedure, consisting in first smoothing and then carrying out a functional principal component analysis. The idea underlying this procedure is that functional data are well approximated by smooth functions, and that rough variations are due to noise. However, it may very well happen that localised features are rough at a global scale but still smooth at some finer scale. In this thesis we put forward a new statistical approach for functional data arising as the sum of two uncorrelated components: one smooth plus one rough. We give non-parametric conditions under which the covariance operators of the smooth and of the rough components are jointly identifiable on the basis of discretely observed data: the covariance operator corresponding to the smooth component must be of finite rank and have real analytic eigenfunctions, while the one corresponding to the rough component must have a banded covariance function. We construct consistent estimators of both covariance operators without assuming knowledge of the true rank or bandwidth. We then use them to estimate the best linear predictors of the the smooth and the rough components of each functional datum. In both the identifiability and the inference part, we do not follow the usual strategy used in functional data analysis which is to first employ smoothing and work with continuous estimate of the covariance operator. Instead, we work directly with the covariance matrix of the discretely observed data, which allows us to use results and tools from linear algebra. In fact, we show that the whole problem of uniquely recovering the covariance operator of the smooth component given the one of the raw data can be seen as a low-rank matrix completion problem, and we make great use of a classical relation between the rank and the minors of a matrix to solve this matrix completion problem. The finite-sample performance of our approach is studied by means of simulation study.

EPFL2017

Chargement

Functional Data Analysis By Matrix Completion1

Marie-Hélène Descary, Victor Panaretos

Functional data analyses typically proceed by smoothing, followed by functional PCA. This paradigm implicitly assumes that rough variation is due to nuisance noise. Nevertheless, relevant functional features such as time-localised or short scale fluctuations may indeed be rough relative to the global scale, but still smooth at shorter scales. These may be confounded with the global smooth components of variation by the smoothing and PCA, potentially distorting the parsimony and interpretability of the analysis. The goal of this paper is to investigate how both smooth and rough variations can be recovered on the basis of discretely observed functional data. Assuming that a functional datum arises as the sum of two uncorrelated components, one smooth and one rough, we develop identifiability conditions for the recovery of the two corresponding covariance operators. The key insight is that they should possess complementary forms of parsimony: one smooth and finite rank (large scale), and the other banded and potentially infinite rank (small scale). Our conditions elucidate the precise interplay between rank, bandwidth and grid resolution. Under these conditions, we show that the recovery problem is equivalent to rank-constrained matrix completion, and exploit this to construct estimators of the two covariances, without assuming knowledge of the true bandwidth or rank; we study their asymptotic behaviour, and then use them to recover the smooth and rough components of each functional datum by best linear prediction. As a result, we effectively produce separate functional PCAs for smooth and rough variation.

INST MATHEMATICAL STATISTICS2019

Chargement

From data to structures

Vasilis Kalofolias

Weighted undirected graphs are a simple, yet powerful way to encode structure in data. A first question we need to address regarding such graphs is how to use them effectively to enhance machine learning problems. A second but more important question is how to obtain such a graph directly from the data. These two questions lead the direction of this thesis. We first explore the use of graphs in matrix completion, with direct implications in recommendation systems. In such systems side information graphs arise naturally, by comparing user behavior and interactions, and by product profiling. Given incomplete object ratings by users, and known graphs for user and object similarities, we propose a new matrix completion model that exploits the graphs to achieve better recommendations. Our model is convex and is solved efficiently with an Alternating Direction Multipliers method (ADMM). It achieves better recommendations than standard convex matrix completion, especially when very few ratings are known a priori, while it is robust to the graph quality. Continuing with the second question, we are interested in learning a graph from data. Data is usually assumed to be smooth on a graph, an assumption associated with graph sparsity and connectivity. Using this assumption, we provide a general graph learning framework that is convex and easy to optimize. We classify graph learning according to the prior structure information in edge-local, node-local or global. Edge-local models are the cheapest computationally, learning each edge independently. Exponential decay graphs and

\epsilon

-neighborhood graphs, the most prevalent weighting schemes in the literature, are such models of our framework. We provide a new model that interpolates between these two, and achieves better performance in many settings. Node-local models take into account the neighborhood of each edge. While they require iterative algorithms, they are stronger models achieving better results. We provide a new node-local model for general graph learning under smoothness assumptions, that redefines the state of the art. We discuss the previous state of the art, two global models of our framework. We provide efficient primal-dual based algorithms for efficiently solving our model and the most prevalent of the previous ones. Our model performs best in many settings. Large scale applications require graphs that are easy to compute, with cost that scales gracefully with respect to the number of nodes, and do not need parameter tuning. We provide the first large scale graph learning solution, based on approximate nearest neighbors with cost of

\mathcal{O}(m\log(m))

for

m

nodes. Our solution automatically finds the needed parameters given the sparsity level needed. We finally explore learning graphs that change in time. In many applications graphs are not static, and data comes from different versions of them. In such cases there is no one graph that correctly summarizes connectivity, and we need to learn several graphs, one for each selected time window. Assuming that graphs change slowly, we provide two new models based on a known global and our own node-local one. Our convex time-varying-graph models can use any convex differentiable loss function for time-change penalization, have a cost linear to the number of time windows, and are easy to parallelize. The obtained graphs fit the data better and provide a good compromise between detail in time and computational cost.

EPFL2016

Chargement

EPFL2017

From data to structures

Vasilis Kalofolias

\epsilon

\mathcal{O}(m\log(m))

for

m

EPFL2016