1,451 research outputs found
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Algebraic Geometry of Matrix Product States
We quantify the representational power of matrix product states (MPS) for
entangled qubit systems by giving polynomial expressions in a pure quantum
state's amplitudes which hold if and only if the state is a translation
invariant matrix product state or a limit of such states. For systems with few
qubits, we give these equations explicitly, considering both periodic and open
boundary conditions. Using the classical theory of trace varieties and trace
algebras, we explain the relationship between MPS and hidden Markov models and
exploit this relationship to derive useful parameterizations of MPS. We make
four conjectures on the identifiability of MPS parameters
When Does a Mixture of Products Contain a Product of Mixtures?
We derive relations between theoretical properties of restricted Boltzmann
machines (RBMs), popular machine learning models which form the building blocks
of deep learning models, and several natural notions from discrete mathematics
and convex geometry. We give implications and equivalences relating
RBM-representable probability distributions, perfectly reconstructible inputs,
Hamming modes, zonotopes and zonosets, point configurations in hyperplane
arrangements, linear threshold codes, and multi-covering numbers of hypercubes.
As a motivating application, we prove results on the relative representational
power of mixtures of product distributions and products of mixtures of pairs of
product distributions (RBMs) that formally justify widely held intuitions about
distributed representations. In particular, we show that a mixture of products
requiring an exponentially larger number of parameters is needed to represent
the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table
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