Measuring the relationship between any pair of variables is a rich and active area of research that is central to scientific practice. In contrast, characterizing the common information among any group of variables is typically a theoretical exercise with few practical methods for high-dimensional data. A promising solution would be a multivariate generalization of the famous Wyner common information, but this approach relies on solving an apparently intractable optimization problem. We leverage the recently introduced information sieve decomposition to formulate an incremental version of the common information problem that admits a simple fixed point solution, fast convergence, and complexity that is linear in the number of variables. This scalable approach allows us to demonstrate the usefulness of common information in high-dimensional learning problems. The sieve outperforms standard methods on dimensionality reduction tasks, solves a blind source separation problem that cannot be solved with ICA, and accurately recovers structure in brain imaging data.
Although the sieve is linear, the information objective that is optimized is nonlinear so the sieve substantially differs from methods like PCA. Superficially, the sieve might seem related to methods like Canonical Correlation Analysis (CCA) that seek to find a @math that makes @math and @math independent, but that method requires some set of labels, @math . One possibility would be to make @math a copy of @math , so that @math is reducing dependence between @math and a copy of itself @cite_18 . However, this objective differs from common information as can be seen by considering the case where @math consists of independent variables. In that case the common information within @math is zero, but @math and its copy still have dependence. The concept of common information'' has largely remained restricted to information-theoretic contexts @cite_17 @cite_24 @cite_7 @cite_29 @cite_11 . The common information in @math that is some variable, @math , is called intersection information and is also an active area of research @cite_25 .
@cite_18 @cite_11 @cite_7 @cite_29 @cite_24 @cite_25 @cite_17
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Let @math be a smooth scheme over an algebraically closed field @math of characteristic zero and @math a regular function, and write @math Crit @math , as a closed subscheme of @math . The motivic vanishing cycle @math is an element of the @math -equivariant motivic Grothendieck ring @math defined by Denef and Loeser math.AG 0006050 and Looijenga math.AG 0006220, and used in Kontsevich and Soibelman's theory of motivic Donaldson-Thomas invariants, arXiv:0811.2435. We prove three main results: (a) @math depends only on the third-order thickenings @math of @math . (b) If @math is another smooth scheme, @math is regular, @math Crit @math , and @math is an embedding with @math and @math an isomorphism, then @math equals @math "twisted" by a motive associated to a principal @math -bundle defined using @math , where now we work in a quotient ring @math of @math . (c) If @math is an "oriented algebraic d-critical locus" in the sense of Joyce arXiv:1304.4508, there is a natural motive @math , such that if @math is locally modelled on Crit @math , then @math is locally modelled on @math . Using results from arXiv:1305.6302, these imply the existence of natural motives on moduli schemes of coherent sheaves on a Calabi-Yau 3-fold equipped with "orientation data", as required in Kontsevich and Soibelman's motivic Donaldson-Thomas theory arXiv:0811.2435, and on intersections of oriented Lagrangians in an algebraic symplectic manifold. This paper is an analogue for motives of results on perverse sheaves of vanishing cycles proved in arXiv:1211.3259. We extend this paper to Artin stacks in arXiv:1312.0090. Measuring the relationship between any pair of variables is a rich and active area of research that is central to scientific practice. In contrast, characterizing the common information among any group of variables is typically a theoretical exercise with few practical methods for high-dimensional data. A promising solution would be a multivariate generalization of the famous Wyner common information, but this approach relies on solving an apparently intractable optimization problem. We leverage the recently introduced information sieve decomposition to formulate an incremental version of the common information problem that admits a simple fixed point solution, fast convergence, and complexity that is linear in the number of variables. This scalable approach allows us to demonstrate the usefulness of common information in high-dimensional learning problems. The sieve outperforms standard methods on dimensionality reduction tasks, solves a blind source separation problem that cannot be solved with ICA, and accurately recovers structure in brain imaging data. We consider the problem of private information retrieval (PIR) from @math non-colluding and replicated databases when the user is equipped with a cache that holds an uncoded fraction @math from each of the @math stored messages in the databases. We assume that the databases are unaware of the cache content. We investigate @math the optimal download cost normalized with the message size as a function of @math , @math , @math . For a fixed @math , @math , we develop an inner bound (converse bound) for the @math curve. The inner bound is a piece-wise linear function in @math that consists of @math line segments. For the achievability, we develop explicit schemes that exploit the cached bits as side information to achieve @math non-degenerate corner points. These corner points differ in the number of cached bits that are used to generate one side information equation. We obtain an outer bound (achievability) for any caching ratio by memory-sharing between these corner points. Thus, the outer bound is also a piece-wise linear function in @math that consists of @math line segments. The inner and the outer bounds match in general for the cases of very low caching ratio ( @math ) and very high caching ratio ( @math ). As a corollary, we fully characterize the optimal download cost caching ratio tradeoff for @math . For general @math , @math , and @math , we show that the largest gap between the achievability and the converse bounds is @math . Our results show that the download cost can be reduced beyond memory-sharing if the databases are unaware of the cached content. Classical methods such as Principal Component Analysis (PCA) and Canonical Correlation Analysis (CCA) are ubiquitous in statistics. However, these techniques are only able to reveal linear relationships in data. Although nonlinear variants of PCA and CCA have been proposed, these are computationally prohibitive in the large scale. In a separate strand of recent research, randomized methods have been proposed to construct features that help reveal nonlinear patterns in data. For basic tasks such as regression or classification, random features exhibit little or no loss in performance, while achieving drastic savings in computational requirements. In this paper we leverage randomness to design scalable new variants of nonlinear PCA and CCA; our ideas extend to key multivariate analysis tools such as spectral clustering or LDA. We demonstrate our algorithms through experiments on real-world data, on which we compare against the state-of-the-art. A simple R implementation of the presented algorithms is provided.