5,293 research outputs found
Harmonic functions on multiplicative graphs and interpolation polynomials
We construct examples of nonnegative harmonic functions on certain graded
graphs: the Young lattice and its generalizations. Such functions first emerged
in harmonic analysis on the infinite symmetric group. Our method relies on
multivariate interpolation polynomials associated with Schur's S and P
functions and with Jack symmetric functions. As a by-product, we compute
certain Selberg-type integrals.Comment: AMSTeX, 35 page
Driven by Compression Progress: A Simple Principle Explains Essential Aspects of Subjective Beauty, Novelty, Surprise, Interestingness, Attention, Curiosity, Creativity, Art, Science, Music, Jokes
I argue that data becomes temporarily interesting by itself to some
self-improving, but computationally limited, subjective observer once he learns
to predict or compress the data in a better way, thus making it subjectively
simpler and more beautiful. Curiosity is the desire to create or discover more
non-random, non-arbitrary, regular data that is novel and surprising not in the
traditional sense of Boltzmann and Shannon but in the sense that it allows for
compression progress because its regularity was not yet known. This drive
maximizes interestingness, the first derivative of subjective beauty or
compressibility, that is, the steepness of the learning curve. It motivates
exploring infants, pure mathematicians, composers, artists, dancers, comedians,
yourself, and (since 1990) artificial systems.Comment: 35 pages, 3 figures, based on KES 2008 keynote and ALT 2007 / DS 2007
joint invited lectur
Deep Unsupervised Learning using Nonequilibrium Thermodynamics
A central problem in machine learning involves modeling complex data-sets
using highly flexible families of probability distributions in which learning,
sampling, inference, and evaluation are still analytically or computationally
tractable. Here, we develop an approach that simultaneously achieves both
flexibility and tractability. The essential idea, inspired by non-equilibrium
statistical physics, is to systematically and slowly destroy structure in a
data distribution through an iterative forward diffusion process. We then learn
a reverse diffusion process that restores structure in data, yielding a highly
flexible and tractable generative model of the data. This approach allows us to
rapidly learn, sample from, and evaluate probabilities in deep generative
models with thousands of layers or time steps, as well as to compute
conditional and posterior probabilities under the learned model. We
additionally release an open source reference implementation of the algorithm
Probabilistic Modeling Paradigms for Audio Source Separation
This is the author's final version of the article, first published as E. Vincent, M. G. Jafari, S. A. Abdallah, M. D. Plumbley, M. E. Davies. Probabilistic Modeling Paradigms for Audio Source Separation. In W. Wang (Ed), Machine Audition: Principles, Algorithms and Systems. Chapter 7, pp. 162-185. IGI Global, 2011. ISBN 978-1-61520-919-4. DOI: 10.4018/978-1-61520-919-4.ch007file: VincentJafariAbdallahPD11-probabilistic.pdf:v\VincentJafariAbdallahPD11-probabilistic.pdf:PDF owner: markp timestamp: 2011.02.04file: VincentJafariAbdallahPD11-probabilistic.pdf:v\VincentJafariAbdallahPD11-probabilistic.pdf:PDF owner: markp timestamp: 2011.02.04Most sound scenes result from the superposition of several sources, which can be separately perceived and analyzed by human listeners. Source separation aims to provide machine listeners with similar skills by extracting the sounds of individual sources from a given scene. Existing separation systems operate either by emulating the human auditory system or by inferring the parameters of probabilistic sound models. In this chapter, the authors focus on the latter approach and provide a joint overview of established and recent models, including independent component analysis, local time-frequency models and spectral template-based models. They show that most models are instances of one of the following two general paradigms: linear modeling or variance modeling. They compare the merits of either paradigm and report objective performance figures. They also,conclude by discussing promising combinations of probabilistic priors and inference algorithms that could form the basis of future state-of-the-art systems
Learning hard quantum distributions with variational autoencoders
Studying general quantum many-body systems is one of the major challenges in
modern physics because it requires an amount of computational resources that
scales exponentially with the size of the system.Simulating the evolution of a
state, or even storing its description, rapidly becomes intractable for exact
classical algorithms. Recently, machine learning techniques, in the form of
restricted Boltzmann machines, have been proposed as a way to efficiently
represent certain quantum states with applications in state tomography and
ground state estimation. Here, we introduce a new representation of states
based on variational autoencoders. Variational autoencoders are a type of
generative model in the form of a neural network. We probe the power of this
representation by encoding probability distributions associated with states
from different classes. Our simulations show that deep networks give a better
representation for states that are hard to sample from, while providing no
benefit for random states. This suggests that the probability distributions
associated to hard quantum states might have a compositional structure that can
be exploited by layered neural networks. Specifically, we consider the
learnability of a class of quantum states introduced by Fefferman and Umans.
Such states are provably hard to sample for classical computers, but not for
quantum ones, under plausible computational complexity assumptions. The good
level of compression achieved for hard states suggests these methods can be
suitable for characterising states of the size expected in first generation
quantum hardware.Comment: v2: 9 pages, 3 figures, journal version with major edits with respect
to v1 (rewriting of section "hard and easy quantum states", extended
discussion on comparison with tensor networks
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