Low-rank tensor decomposition generalizes low-rank matrix approximation and is a powerful technique for discovering low-dimensional structure in high-dimensional data. In this paper, we study Tucker decompositions and use tools from randomized numerical linear algebra called ridge leverage scores to accelerate the core tensor update step in the widely-used alternating least squares (ALS) algorithm. Updating the core tensor, a severe bottleneck in ALS, is a highly-structured ridge regression problem where the design matrix is a Kronecker product of the factor matrices. We show how to use approximate ridge leverage scores to construct a sketched instance for any ridge regression problem such that the solution vector for the sketched problem is a $(1+\varepsilon)$-approximation to the original instance. Moreover, we show that classical leverage scores suffice as an approximation, which then allows us to exploit the Kronecker structure and update the core tensor in time that depends predominantly on the rank and the sketching parameters (i.e., sublinear in the size of the input tensor). We also give upper bounds for ridge leverage scores as rows are removed from the design matrix (e.g., if the tensor has missing entries), and we demonstrate the effectiveness of our approximate ridge regressioni algorithm for large, low-rank Tucker decompositions on both synthetic and real-world data.
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There are a variety of settings where vague prior information may be available on the importance of predictors in high-dimensional regression settings. Examples include ordering on the variables offered by their empirical variances (which is typically discarded through standardisation), the lag of predictors when fitting autoregressive models in time series settings, or the level of missingness of the variables. Whilst such orderings may not match the true importance of variables, we argue that there is little to be lost, and potentially much to be gained, by using them. We propose a simple scheme involving fitting a sequence of models indicated by the ordering. We show that the computational cost for fitting all models when ridge regression is used is no more than for a single fit of ridge regression, and describe a strategy for Lasso regression that makes use of previous fits to greatly speed up fitting the entire sequence of models. We propose to select a final estimator by cross-validation and provide a general result on the quality of the best performing estimator on a test set selected from among a number $M$ of competing estimators in a high-dimensional linear regression setting. Our result requires no sparsity assumptions and shows that only a $\log M$ price is incurred compared to the unknown best estimator. We demonstrate the effectiveness of our approach when applied to missing or corrupted data, and time series settings. An R package is available on github.
Low-rank matrix approximation is one of the central concepts in machine learning, with applications in dimension reduction, de-noising, multivariate statistical methodology, and many more. A recent extension to LRMA is called low-rank matrix completion (LRMC). It solves the LRMA problem when some observations are missing and is especially useful for recommender systems. In this paper, we consider an element-wise weighted generalization of LRMA. The resulting weighted low-rank matrix approximation technique therefore covers LRMC as a special case with binary weights. WLRMA has many applications. For example, it is an essential component of GLM optimization algorithms, where an exponential family is used to model the entries of a matrix, and the matrix of natural parameters admits a low-rank structure. We propose an algorithm for solving the weighted problem, as well as two acceleration techniques. Further, we develop a non-SVD modification of the proposed algorithm that is able to handle extremely high-dimensional data. We compare the performance of all the methods on a small simulation example as well as a real-data application.
Several methods of triclustering of three dimensional data require the specification of the cluster size in each dimension. This introduces a certain degree of arbitrariness. To address this issue, we propose a new method, namely the multi-slice clustering (MSC) for a 3-order tensor data set. We analyse, in each dimension or tensor mode, the spectral decomposition of each tensor slice, i.e. a matrix. Thus, we define a similarity measure between matrix slices up to a threshold (precision) parameter, and from that, identify a cluster. The intersection of all partial clusters provides the desired triclustering. The effectiveness of our algorithm is shown on both synthetic and real-world data sets.
We study the theoretical properties of random Fourier features classification with Lipschitz continuous loss functions such as support vector machine and logistic regression. Utilizing the regularity condition, we show for the first time that random Fourier features classification can achieve $O(1/\sqrt{n})$ learning rate with only $\Omega(\sqrt{n} \log n)$ features, as opposed to $\Omega(n)$ features suggested by previous results. Our study covers the standard feature sampling method for which we reduce the number of features required, as well as a problem-dependent sampling method which further reduces the number of features while still keeping the optimal generalization property. Moreover, we prove that the random Fourier features classification can obtain a fast $O(1/n)$ learning rate for both sampling schemes under Massart's low noise assumption. Our results demonstrate the potential effectiveness of random Fourier features approximation in reducing the computational complexity (roughly from $O(n^3)$ in time and $O(n^2)$ in space to $O(n^2)$ and $O(n\sqrt{n})$ respectively) without having to trade-off the statistical prediction accuracy. In addition, the achieved trade-off in our analysis is at least the same as the optimal results in the literature under the worst case scenario and significantly improves the optimal results under benign regularity conditions.
We propose a general and scalable approximate sampling strategy for probabilistic models with discrete variables. Our approach uses gradients of the likelihood function with respect to its discrete inputs to propose updates in a Metropolis-Hastings sampler. We show empirically that this approach outperforms generic samplers in a number of difficult settings including Ising models, Potts models, restricted Boltzmann machines, and factorial hidden Markov models. We also demonstrate the use of our improved sampler for training deep energy-based models on high dimensional discrete data. This approach outperforms variational auto-encoders and existing energy-based models. Finally, we give bounds showing that our approach is near-optimal in the class of samplers which propose local updates.
Several recent applications of optimal transport (OT) theory to machine learning have relied on regularization, notably entropy and the Sinkhorn algorithm. Because matrix-vector products are pervasive in the Sinkhorn algorithm, several works have proposed to \textit{approximate} kernel matrices appearing in its iterations using low-rank factors. Another route lies instead in imposing low-rank constraints on the feasible set of couplings considered in OT problems, with no approximations on cost nor kernel matrices. This route was first explored by Forrow et al., 2018, who proposed an algorithm tailored for the squared Euclidean ground cost, using a proxy objective that can be solved through the machinery of regularized 2-Wasserstein barycenters. Building on this, we introduce in this work a generic approach that aims at solving, in full generality, the OT problem under low-rank constraints with arbitrary costs. Our algorithm relies on an explicit factorization of low rank couplings as a product of \textit{sub-coupling} factors linked by a common marginal; similar to an NMF approach, we alternatively updates these factors. We prove the non-asymptotic stationary convergence of this algorithm and illustrate its efficiency on benchmark experiments.
Leveraging biased click data for optimizing learning to rank systems has been a popular approach in information retrieval. Because click data is often noisy and biased, a variety of methods have been proposed to construct unbiased learning to rank (ULTR) algorithms for the learning of unbiased ranking models. Among them, automatic unbiased learning to rank (AutoULTR) algorithms that jointly learn user bias models (i.e., propensity models) with unbiased rankers have received a lot of attention due to their superior performance and low deployment cost in practice. Despite their differences in theories and algorithm design, existing studies on ULTR usually use uni-variate ranking functions to score each document or result independently. On the other hand, recent advances in context-aware learning-to-rank models have shown that multivariate scoring functions, which read multiple documents together and predict their ranking scores jointly, are more powerful than uni-variate ranking functions in ranking tasks with human-annotated relevance labels. Whether such superior performance would hold in ULTR with noisy data, however, is mostly unknown. In this paper, we investigate existing multivariate scoring functions and AutoULTR algorithms in theory and prove that permutation invariance is a crucial factor that determines whether a context-aware learning-to-rank model could be applied to existing AutoULTR framework. Our experiments with synthetic clicks on two large-scale benchmark datasets show that AutoULTR models with permutation-invariant multivariate scoring functions significantly outperform those with uni-variate scoring functions and permutation-variant multivariate scoring functions.
Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.
Recent studies show that large-scale sketch-based image retrieval (SBIR) can be efficiently tackled by cross-modal binary representation learning methods, where Hamming distance matching significantly speeds up the process of similarity search. Providing training and test data subjected to a fixed set of pre-defined categories, the cutting-edge SBIR and cross-modal hashing works obtain acceptable retrieval performance. However, most of the existing methods fail when the categories of query sketches have never been seen during training. In this paper, the above problem is briefed as a novel but realistic zero-shot SBIR hashing task. We elaborate the challenges of this special task and accordingly propose a zero-shot sketch-image hashing (ZSIH) model. An end-to-end three-network architecture is built, two of which are treated as the binary encoders. The third network mitigates the sketch-image heterogeneity and enhances the semantic relations among data by utilizing the Kronecker fusion layer and graph convolution, respectively. As an important part of ZSIH, we formulate a generative hashing scheme in reconstructing semantic knowledge representations for zero-shot retrieval. To the best of our knowledge, ZSIH is the first zero-shot hashing work suitable for SBIR and cross-modal search. Comprehensive experiments are conducted on two extended datasets, i.e., Sketchy and TU-Berlin with a novel zero-shot train-test split. The proposed model remarkably outperforms related works.
This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.
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