arXiv subjects

Mark Tygert

Publications and source records attributed to Mark Tygert.

At least 37 records · Page 2Linked to original sources

arXiv2018

Regression-aware decompositions

Linear least-squares regression with a "design" matrix A approximates a given matrix B via minimization of the spectral- or Frobenius-norm discrepancy ||AX-B|| over every conformingly sized matrix X. Another popular approximation is low-rank approximation via principal component analysis (PCA) -- which is essentially singular value decomposition (SVD) -- or interpolative decomposition (ID). Classically, PCA/SVD and ID operate solely with the matrix B being approximated, not supervised by any auxiliary matrix A. However, linear least-squares regression models can inform the ID, yielding regression-aware ID. As a bonus, this provides an interpretation as regression-aware PCA for a kind of canonical correlation analysis between A and B. The regression-aware decompositions effectively enable supervision to inform classical dimensionality reduction, which classically has been totally unsupervised. The regression-aware decompositions reveal the structure inherent in B that is relevant to regression against A.

stat.ME↗

arXiv2018

Randomized algorithms for distributed computation of principal component analysis and singular value decomposition

Randomized algorithms provide solutions to two ubiquitous problems: (1) the distributed calculation of a principal component analysis or singular value decomposition of a highly rectangular matrix, and (2) the distributed calculation of a low-rank approximation (in the form of a singular value decomposition) to an arbitrary matrix. Carefully honed algorithms yield results that are uniformly superior to those of the stock, deterministic implementations in Spark (the popular platform for distributed computation); in particular, whereas the stock software will without warning return left singular vectors that are far from numerically orthonormal, a significantly burnished randomized implementation generates left singular vectors that are numerically orthonormal to nearly the machine precision.

cs.DC↗

arXiv2016

Accurate principal component analysis via a few iterations of alternating least squares

A few iterations of alternating least squares with a random starting point provably suffice to produce nearly optimal spectral- and Frobenius-norm accuracies of low-rank approximations to a matrix; iterating to convergence is unnecessary. Thus, software implementing alternating least squares can be retrofitted via appropriate setting of parameters to calculate nearly optimally accurate low-rank approximations highly efficiently, with no need for convergence.

math.NA↗

arXiv2016

Convolutional networks and learning invariant to homogeneous multiplicative scalings

The conventional classification schemes -- notably multinomial logistic regression -- used in conjunction with convolutional networks (convnets) are classical in statistics, designed without consideration for the usual coupling with convnets, stochastic gradient descent, and backpropagation. In the specific application to supervised learning for convnets, a simple scale-invariant classification stage turns out to be more robust than multinomial logistic regression, appears to result in slightly lower errors on several standard test sets, has similar computational costs, and features precise control over the actual rate of learning. "Scale-invariant" means that multiplying the input values by any nonzero scalar leaves the output unchanged.

cs.LG↗

arXiv2016

Poor starting points in machine learning

Poor (even random) starting points for learning/training/optimization are common in machine learning. In many settings, the method of Robbins and Monro (online stochastic gradient descent) is known to be optimal for good starting points, but may not be optimal for poor starting points -- indeed, for poor starting points Nesterov acceleration can help during the initial iterations, even though Nesterov methods not designed for stochastic approximation could hurt during later iterations. The common practice of training with nontrivial minibatches enhances the advantage of Nesterov acceleration.

cs.LG↗

arXiv2015

A mathematical motivation for complex-valued convolutional networks

A complex-valued convolutional network (convnet) implements the repeated application of the following composition of three operations, recursively applying the composition to an input vector of nonnegative real numbers: (1) convolution with complex-valued vectors followed by (2) taking the absolute value of every entry of the resulting vectors followed by (3) local averaging. For processing real-valued random vectors, complex-valued convnets can be viewed as "data-driven multiscale windowed power spectra," "data-driven multiscale windowed absolute spectra," "data-driven multiwavelet absolute values," or (in their most general configuration) "data-driven nonlinear multiwavelet packets." Indeed, complex-valued convnets can calculate multiscale windowed spectra when the convnet filters are windowed complex-valued exponentials. Standard real-valued convnets, using rectified linear units (ReLUs), sigmoidal (for example, logistic or tanh) nonlinearities, max. pooling, etc., do not obviously exhibit the same exact correspondence with data-driven wavelets (whereas for complex-valued convnets, the correspondence is much more than just a vague analogy). Courtesy of the exact correspondence, the remarkably rich and rigorous body of mathematical analysis for wavelets applies directly to (complex-valued) convnets.

cs.LG↗

arXiv2014

An implementation of a randomized algorithm for principal component analysis

Recent years have witnessed intense development of randomized methods for low-rank approximation. These methods target principal component analysis (PCA) and the calculation of truncated singular value decompositions (SVD). The present paper presents an essentially black-box, fool-proof implementation for Mathworks' MATLAB, a popular software platform for numerical computation. As illustrated via several tests, the randomized algorithms for low-rank approximation outperform or at least match the classical techniques (such as Lanczos iterations) in basically all respects: accuracy, computational efficiency (both speed and memory usage), ease-of-use, parallelizability, and reliability. However, the classical procedures remain the methods of choice for estimating spectral norms, and are far superior for calculating the least singular values and corresponding singular vectors (or singular subspaces).

stat.CO↗

arXiv2013

Testing goodness-of-fit for logistic regression

Explicitly accounting for all applicable independent variables, even when the model being tested does not, is critical in testing goodness-of-fit for logistic regression. This can increase statistical power by orders of magnitude.

stat.ME↗

arXiv2013

Significance testing without truth

A popular approach to significance testing proposes to decide whether the given hypothesized statistical model is likely to be true (or false). Statistical decision theory provides a basis for this approach by requiring every significance test to make a decision about the truth of the hypothesis/model under consideration. Unfortunately, many interesting and useful models are obviously false (that is, not exactly true) even before considering any data. Fortunately, in practice a significance test need only gauge the consistency (or inconsistency) of the observed data with the assumed hypothesis/model -- without enquiring as to whether the assumption is likely to be true (or false), or whether some alternative is likely to be true (or false). In this practical formulation, a significance test rejects a hypothesis/model only if the observed data is highly improbable when calculating the probability while assuming the hypothesis being tested; the significance test only gauges whether the observed data likely invalidates the assumed hypothesis, and cannot decide that the assumption -- however unmistakably false -- is likely to be false a priori, without any data.

stat.ME↗

arXiv2012

A comparison of the discrete Kolmogorov-Smirnov statistic and the Euclidean distance

Goodness-of-fit tests gauge whether a given set of observations is consistent (up to expected random fluctuations) with arising as independent and identically distributed (i.i.d.) draws from a user-specified probability distribution known as the "model." The standard gauges involve the discrepancy between the model and the empirical distribution of the observed draws. Some measures of discrepancy are cumulative; others are not. The most popular cumulative measure is the Kolmogorov-Smirnov statistic; when all probability distributions under consideration are discrete, a natural noncumulative measure is the Euclidean distance between the model and the empirical distributions. In the present paper, both mathematical analysis and its illustration via various data sets indicate that the Kolmogorov-Smirnov statistic tends to be more powerful than the Euclidean distance when there is a natural ordering for the values that the draws can take -- that is, when the data is ordinal -- whereas the Euclidean distance is more reliable and more easily understood than the Kolmogorov-Smirnov statistic when there is no natural ordering (or partial order) -- that is, when the data is nominal.

stat.ME↗

arXiv2012

Computing the asymptotic power of a Euclidean-distance test for goodness-of-fit

A natural (yet unconventional) test for goodness-of-fit measures the discrepancy between the model and empirical distributions via their Euclidean distance (or, equivalently, via its square). The present paper characterizes the statistical power of such a test against a family of alternative distributions, in the limit that the number of observations is large, with every alternative departing from the model in the same direction. Specifically, the paper provides an efficient numerical method for evaluating the cumulative distribution function (cdf) of the square of the Euclidean distance between the model and empirical distributions under the alternatives, in the limit that the number of observations is large. The paper illustrates the scheme by plotting the asymptotic power (as a function of the significance level) for several examples.

stat.CO↗

arXiv2012

An introduction to how chi-square and classical exact tests often wildly misreport significance and how the remedy lies in computers

Goodness-of-fit tests based on the Euclidean distance often outperform chi-square and other classical tests (including the standard exact tests) by at least an order of magnitude when the model being tested for goodness-of-fit is a discrete probability distribution that is not close to uniform. The present article discusses numerous examples of this. Goodness-of-fit tests based on the Euclidean metric are now practical and convenient: although the actual values taken by the Euclidean distance and similar goodness-of-fit statistics are seldom humanly interpretable, black-box computer programs can rapidly calculate their precise significance.

stat.ME↗

arXiv2012

Testing the significance of assuming homogeneity in contingency-tables/cross-tabulations

The model for homogeneity of proportions in a two-way contingency-table/cross-tabulation is the same as the model of independence, except that the probabilistic process generating the data is viewed as fixing the column totals (but not the row totals). When gauging the consistency of observed data with the assumption of independence, recent work has illustrated that the Euclidean/Frobenius/Hilbert-Schmidt distance is often far more statistically powerful than the classical statistics such as chi-square, the log-likelihood-ratio (G), the Freeman-Tukey/Hellinger distance, and other members of the Cressie-Read power-divergence family. The present paper indicates that the Euclidean/Frobenius/Hilbert-Schmidt distance can be more powerful for gauging the consistency of observed data with the assumption of homogeneity, too.

stat.ME↗

arXiv2011

Computing the confidence levels for a root-mean-square test of goodness-of-fit, II

This paper extends our earlier article, "Computing the confidence levels for a root-mean-square test of goodness-of-fit;" unlike in the earlier article, the models in the present paper involve parameter estimation -- both the null and alternative hypotheses in the associated tests are composite. We provide efficient black-box algorithms for calculating the asymptotic confidence levels of a variant on the classic chi-squared test. In some circumstances, it is also feasible to compute the exact confidence levels via Monte Carlo simulation.

stat.CO↗

arXiv2011

Chi-square and classical exact tests often wildly misreport significance; the remedy lies in computers

If a discrete probability distribution in a model being tested for goodness-of-fit is not close to uniform, then forming the Pearson chi-square statistic can involve division by nearly zero. This often leads to serious trouble in practice -- even in the absence of round-off errors -- as the present article illustrates via numerous examples. Fortunately, with the now widespread availability of computers, avoiding all the trouble is simple and easy: without the problematic division by nearly zero, the actual values taken by goodness-of-fit statistics are not humanly interpretable, but black-box computer programs can rapidly calculate their precise significance.

stat.ME↗

arXiv2011

An algorithm for the principal component analysis of large data sets

Recently popularized randomized methods for principal component analysis (PCA) efficiently and reliably produce nearly optimal accuracy --- even on parallel processors --- unlike the classical (deterministic) alternatives. We adapt one of these randomized methods for use with data sets that are too large to be stored in random-access memory (RAM). (The traditional terminology is that our procedure works efficiently "out-of-core.") We illustrate the performance of the algorithm via several numerical examples. For example, we report on the PCA of a data set stored on disk that is so large that less than a hundredth of it can fit in our computer's RAM.

stat.CO↗

arXiv2011

Computing the confidence levels for a root-mean-square test of goodness-of-fit

The classic chi-squared statistic for testing goodness-of-fit has long been a cornerstone of modern statistical practice. The statistic consists of a sum in which each summand involves division by the probability associated with the corresponding bin in the distribution being tested for goodness-of-fit. Typically this division should precipitate rebinning to uniformize the probabilities associated with the bins, in order to make the test reasonably powerful. With the now widespread availability of computers, there is no longer any need for this. The present paper provides efficient black-box algorithms for calculating the asymptotic confidence levels of a variant on the classic chi-squared test which omits the problematic division. In many circumstances, it is also feasible to compute the exact confidence levels via Monte Carlo simulation.

stat.CO↗

arXiv2010

Statistical tests for whether a given set of independent, identically distributed draws does not come from a specified probability density

We discuss several tests for whether a given set of independent and identically distributed (i.i.d.) draws does not come from a specified probability density function. The most commonly used are Kolmogorov-Smirnov tests, particularly Kuiper's variant, which focus on discrepancies between the cumulative distribution function for the specified probability density and the empirical cumulative distribution function for the given set of i.i.d. draws. Unfortunately, variations in the probability density function often get smoothed over in the cumulative distribution function, making it difficult to detect discrepancies in regions where the probability density is small in comparison with its values in surrounding regions. We discuss tests without this deficiency, complementing the classical methods. The tests of the present paper are based on the plain fact that it is unlikely to draw a random number whose probability is small, provided that the draw is taken from the same distribution used in calculating the probability (thus, if we draw a random number whose probability is small, then we can be confident that we did not draw the number from the same distribution used in calculating the probability).

stat.ME↗