2,020 research outputs found
Statistical Physics and Representations in Real and Artificial Neural Networks
This document presents the material of two lectures on statistical physics
and neural representations, delivered by one of us (R.M.) at the Fundamental
Problems in Statistical Physics XIV summer school in July 2017. In a first
part, we consider the neural representations of space (maps) in the
hippocampus. We introduce an extension of the Hopfield model, able to store
multiple spatial maps as continuous, finite-dimensional attractors. The phase
diagram and dynamical properties of the model are analyzed. We then show how
spatial representations can be dynamically decoded using an effective Ising
model capturing the correlation structure in the neural data, and compare
applications to data obtained from hippocampal multi-electrode recordings and
by (sub)sampling our attractor model. In a second part, we focus on the problem
of learning data representations in machine learning, in particular with
artificial neural networks. We start by introducing data representations
through some illustrations. We then analyze two important algorithms, Principal
Component Analysis and Restricted Boltzmann Machines, with tools from
statistical physics
A Class of -Invariant Topological Phases of Interacting Electrons
We describe a class of parity- and time-reversal-invariant topological states
of matter which can arise in correlated electron systems in 2+1-dimensions.
These states are characterized by particle-like excitations exhibiting exotic
braiding statistics. and invariance are maintained by a `doubling' of
the low-energy degrees of freedom which occurs naturally without doubling the
underlying microscopic degrees of freedom. The simplest examples have been the
subject of considerable interest as proposed mechanisms for high-
superconductivity. One is the `doubled' version (i.e. two opposite-chirality
copies) of the U(1) chiral spin liquid. The second example corresponds to
gauge theory, which describes a scenario for spin-charge separation. Our main
concern, with an eye towards applications to quantum computation, are richer
models which support non-Abelian statistics. All of these models, richer or
poorer, lie in a tightly-organized discrete family. The physical inference is
that a material manifesting the gauge theory or a doubled chiral spin
liquid might be easily altered to one capable of universal quantum computation.
These phases of matter have a field-theoretic description in terms of gauge
theories which, in their infrared limits, are topological field theories. We
motivate these gauge theories using a parton model or slave-fermion
construction and show how they can be solved exactly. The structure of the
resulting Hilbert spaces can be understood in purely combinatorial terms. The
highly-constrained nature of this combinatorial construction, phrased in the
language of the topology of curves on surfaces, lays the groundwork for a
strategy for constructing microscopic lattice models which give rise to these
phases.Comment: Typos fixed, references adde
Conformal field theories in anti-de Sitter space
In this paper we discuss the dynamics of conformal field theories on anti-de
Sitter space, focussing on the special case of the N=4 supersymmetric
Yang-Mills theory on AdS_4. We argue that the choice of boundary conditions, in
particular for the gauge field, has a large effect on the dynamics. For
example, for weak coupling, one of two natural choices of boundary conditions
for the gauge field leads to a large N deconfinement phase transition as a
function of the temperature, while the other does not. For boundary conditions
that preserve supersymmetry, the strong coupling dynamics can be analyzed using
S-duality (relevant for g_{YM} >> 1), utilizing results of Gaiotto and Witten,
as well as by using the AdS/CFT correspondence (relevant for large N and large
't Hooft coupling). We argue that some very specific choices of boundary
conditions lead to a simple dual gravitational description for this theory,
while for most choices the gravitational dual is not known. In the cases where
the gravitational dual is known, we discuss the phase structure at large 't
Hooft coupling.Comment: 57 pages, 1 figure. v2: fixed typo
Aspects of Type IIB Theory on ALE Spaces
D-brane technology and strong/weak coupling duality supplement traditional
orbifold techniques by making certain background geometries more accessible. In
this spirit, we consider some of the geometric properties of the type IIB
theory on R^6 \times M where M is an `Asymptotically Locally Euclidean (ALE)'
gravitational instanton. Given the self-duality of the theory, we can extract
the geometry (both singular and resolved) seen by the weakly coupled IIB string
by studying the physics of a D1-brane probe. The construction is both amusing
and instructive, as the physics of the probe completely captures the
mathematics of the construction of ALE instantons via `HyperKahler Quotients',
as presented by Kronheimer. This relation has been noted by Douglas and Moore
for the A-series. We extend the explicit construction to the case of the D- and
E-series -- uncovering a quite beautiful structure -- and highlight how all of
the elements of the mathematical construction find their counterparts in the
physics of the type IIB D-string. We discuss the explicit ALE metrics which may
be obtained using these techniques, and comment on the role duality plays in
relating gauged linear sigma models to conformal field theories.Comment: 27 pages, three figures. Uses harvmac.tex and epsf.tex (sentences
corrected on pages 13+14, reference added, small addition to final remarks
Free particles from Brauer algebras in complex matrix models
The gauge invariant degrees of freedom of matrix models based on an N x N
complex matrix, with U(N) gauge symmetry, contain hidden free particle
structures. These are exhibited using triangular matrix variables via the Schur
decomposition. The Brauer algebra basis for complex matrix models developed
earlier is useful in projecting to a sector which matches the state counting of
N free fermions on a circle. The Brauer algebra projection is characterized by
the vanishing of a scale invariant laplacian constructed from the complex
matrix. The special case of N=2 is studied in detail: the ring of gauge
invariant functions as well as a ring of scale and gauge invariant differential
operators are characterized completely. The orthonormal basis of wavefunctions
in this special case is completely characterized by a set of five commuting
Hamiltonians, which display free particle structures. Applications to the
reduced matrix quantum mechanics coming from radial quantization in N=4 SYM are
described. We propose that the string dual of the complex matrix harmonic
oscillator quantum mechanics has an interpretation in terms of strings and
branes in 2+1 dimensions.Comment: 64 pages, v2: Exposition improved, minor corrections; v3: Typos
corrected, published versio
Deep clustering: Discriminative embeddings for segmentation and separation
We address the problem of acoustic source separation in a deep learning
framework we call "deep clustering." Rather than directly estimating signals or
masking functions, we train a deep network to produce spectrogram embeddings
that are discriminative for partition labels given in training data. Previous
deep network approaches provide great advantages in terms of learning power and
speed, but previously it has been unclear how to use them to separate signals
in a class-independent way. In contrast, spectral clustering approaches are
flexible with respect to the classes and number of items to be segmented, but
it has been unclear how to leverage the learning power and speed of deep
networks. To obtain the best of both worlds, we use an objective function that
to train embeddings that yield a low-rank approximation to an ideal pairwise
affinity matrix, in a class-independent way. This avoids the high cost of
spectral factorization and instead produces compact clusters that are amenable
to simple clustering methods. The segmentations are therefore implicitly
encoded in the embeddings, and can be "decoded" by clustering. Preliminary
experiments show that the proposed method can separate speech: when trained on
spectrogram features containing mixtures of two speakers, and tested on
mixtures of a held-out set of speakers, it can infer masking functions that
improve signal quality by around 6dB. We show that the model can generalize to
three-speaker mixtures despite training only on two-speaker mixtures. The
framework can be used without class labels, and therefore has the potential to
be trained on a diverse set of sound types, and to generalize to novel sources.
We hope that future work will lead to segmentation of arbitrary sounds, with
extensions to microphone array methods as well as image segmentation and other
domains.Comment: Originally submitted on June 5, 201
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