Mixtures and products in two graphical models

Seigal, Anna; Montúfar, Guido

doi:10.48550/arxiv.1709.05276

Cited by 3 publications

(11 citation statements)

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“…The constraints on triangulations imposed by the number of peak, as described in Theorem 1, agree with observations from simulations in [30] (Figure 5). Related results have been described in [29].…”

Section: Peak Patternmentioning

confidence: 63%

Geometry of fitness landscapes: Peaks, shapes and universal positive epistasis

Crona¹,

Krug²,

Srivastava³

2021

Preprint

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Darwinian evolution is driven by random mutations, genetic recombination (gene shuffling) and selection that favors genotypes with high fitness. For systems where each genotype can be represented as a bitstring of length L, an overview of possible evolutionary trajectories is provided by the L-cube graph with nodes labeled by genotypes and edges directed toward the genotype with higher fitness. Peaks (sinks in the graphs) are important since a population can get stranded at a suboptimal peak. The fitness landscape is defined by the fitness values of all genotypes in the system. Some notion of curvature is necessary for a more complete analysis of the landscapes, including the effect of recombination. The shape approach uses triangulations (shapes) induced by fitness landscapes. The main topic for this work is the interplay between peak patterns and shapes. Because of constraints on the shapes for L = 3 imposed by peaks there are in total 25 possible combinations of peak patterns and shapes. Similar constraints exist for higher L. Specifically, we show that the constraints induced by the staircase triangulation can be formulated as a condition of universal positive epistasis, an order relation on the fitness effects of arbitrary sets of mutations that respects the inclusion relation between the corresponding genetic backgrounds. We apply the concept to a large protein fitness landscapes for an immunoglobulin-binding protein expressed in Streptococcal bacteria.

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Section: Peak Patternmentioning

confidence: 63%

Geometry of fitness landscapes: Peaks, shapes and universal positive epistasis

Crona¹,

Krug²,

Srivastava³

2021

Preprint

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“…RBMs have been studied intensively, with tools from optimization, algebraic geometry, combinatorics, coding theory, polyhedral geometry, and information geometry among others. Some of the advances over the past few years include results in relation to their approximation properties [77,43,58,57], dimension [17,53,55], semialgebraic description [18,68], efficiency of representation [45,54], sequential optimization [23,26], statistical complexity [10], sampling and training [64,22,23,26], information geometry [7,6,41].…”

Section: Brief Overviewmentioning

confidence: 99%

“…This kind of constraint, coming from constraints on the mixture components, have been studied in [54]. An exact description of the constraints that apply to the probability distributions within RBM 3,2 was obtained recently in [68]. We comment on this later in Section 8.…”

Section: Restricted Mixtures Of Productsmentioning

confidence: 99%

“…The structure of the optimization landscape and critical points of these methods is the subject of current studies. In Section 8 we discuss results from [68] showing that an RBM model can indeed have several local optimizers with different values of the likelihood function, but also that in some cases, the optimization problem may be solvable in closed form.…”

Section: Optimization Landscapementioning

confidence: 99%

“…This allows us to compute the maximum likelihood estimators of a polyhedral family in the following way. A related discussion was conducted recently in [2] in the context of mixtures of products, and in [68] in the context of RBMs.…”

Section: Divergence To Polyhedral Exponential Familiesmentioning

confidence: 99%

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Restricted Boltzmann Machines: Introduction and Review

Montúfar

2018

Springer Proceedings in Mathematics &Amp; Statistics

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The restricted Boltzmann machine is a network of stochastic units with undirected interactions between pairs of visible and hidden units. This model was popularized as a building block of deep learning architectures and has continued to play an important role in applied and theoretical machine learning. Restricted Boltzmann machines carry a rich structure, with connections to geometry, applied algebra, probability, statistics, machine learning, and other areas. The analysis of these models is attractive in its own right and also as a platform to combine and generalize mathematical tools for graphical models with hidden variables. This article gives an introduction to the mathematical analysis of restricted Boltzmann machines, reviews recent results on the geometry of the sets of probability distributions representable by these models, and suggests a few directions for further investigation.

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Conditional independence ideals with hidden variables

Clarke

Mohammadi

Rauh

2020

Advances in Applied Mathematics

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We study a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables. Such CI statements correspond to determinantal conditions on flattenings of marginals of the tensor of joint probabilities of the observed random variables. We focus on an example that generalizes the CI ideals of the intersection axiom. In this example, the minimal primes are again determinantal ideals, which is not true in general.

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Mixtures and products in two graphical models

Cited by 3 publications

References 0 publications

Geometry of fitness landscapes: Peaks, shapes and universal positive epistasis

Geometry of fitness landscapes: Peaks, shapes and universal positive epistasis

Restricted Boltzmann Machines: Introduction and Review

Conditional independence ideals with hidden variables

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