901 research outputs found
Monte Carlo simulations of the three-dimensional XY spin glass focusing on the chiral and the spin order
The ordering of the three-dimensional isotropic {\it XY} spin glass with the
nearest-neighbor random Gaussian coupling is studied by extensive Monte Carlo
simulations. To investigate the ordering of the spin and the chirality, we
compute several independent physical quantities including the glass order
parameter, the Binder parameter, the correlation-length ratio, the overlap
distribution and the non-self-averageness parameter, {\it etc}, for both the
spin-glass (SG) and the chiral-glass (CG) degrees of freedom. Evidence of the
spin-chirality decoupling, {\it i.e.}, the CG and the SG order occurring at two
separated temperatures, , is obtained from the glass order
parameter, which is fully corroborated by the Binder parameter. By contrast,
the CG correlation-length ratio yields a rather pathological and inconsistent
result in the range of sizes we studied, which may originate from the
finite-size effect associated with a significant short-length drop-off of the
spatial CG correlations. Finite-size-scaling analysis yields the CG exponents
and , and the
SG exponents and
. The obtained exponents are close to those of
the Heisenberg SG, but are largely different from those of the Ising SG. The
chiral overlap distribution and the chiral Binder parameter exhibit the feature
of a continuous one-step replica-symmetry breaking (1RSB), consistently with
the previous reports. Such a 1RSB feature is again in common with that of the
Heisenberg SG, but is different from the Ising one, which may be the cause of
the difference in the CG critical properties from the Ising SG ones despite of
a common symmetry.Comment: 15 pages, 31 figure
Zeros of the partition function and dynamical singularities in spin-glass systems
We study spin-glass systems characterized by continuous occurrence of
singularities. The theory of Lee-Yang zeros is used to find the singularities.
By using the replica method in mean-field systems, we show that two-dimensional
distributions of zeros of the partition function in a complex parameter plane
are characteristic feature of random systems. The results of several models
indicate that the concept of chaos in the spin-glass state is different from
that of the replica symmetry breaking. We discuss that a chaotic phase at
imaginary temperature is different from the spin-glass phase and is accessible
by quantum dynamics in a quenching protocol.Comment: 11 pages, 6 figures, proceedings of the ICSG201
Dynamical Singularities of Glassy Systems in a Quantum Quench
We present a prototype of behavior of glassy systems driven by quantum
dynamics in a quenching protocol by analyzing the random energy model in a
transverse field. We calculate several types of dynamical quantum amplitude and
find a freezing transition at some critical time. The behavior is understood by
the partition-function zeros in the complex temperature plane. We discuss the
properties of the freezing phase as a dynamical chaotic phase, which are
contrasted to those of the spin-glass phase in the static system.Comment: 6 pages, 5 figure
Sparse approximation problem: how rapid simulated annealing succeeds and fails
Information processing techniques based on sparseness have been actively
studied in several disciplines. Among them, a mathematical framework to
approximately express a given dataset by a combination of a small number of
basis vectors of an overcomplete basis is termed the {\em sparse
approximation}. In this paper, we apply simulated annealing, a metaheuristic
algorithm for general optimization problems, to sparse approximation in the
situation where the given data have a planted sparse representation and noise
is present. The result in the noiseless case shows that our simulated annealing
works well in a reasonable parameter region: the planted solution is found
fairly rapidly. This is true even in the case where a common relaxation of the
sparse approximation problem, the -relaxation, is ineffective. On the
other hand, when the dimensionality of the data is close to the number of
non-zero components, another metastable state emerges, and our algorithm fails
to find the planted solution. This phenomenon is associated with a first-order
phase transition. In the case of very strong noise, it is no longer meaningful
to search for the planted solution. In this situation, our algorithm determines
a solution with close-to-minimum distortion fairly quickly.Comment: 12 pages, 7 figures, a proceedings of HD^3-201
Partition-function zeros of spherical spin glasses and their relevance to chaos
We investigate partition-function zeros of the many-body interacting
spherical spin glass, the so-called -spin spherical model, with respect to
the complex temperature in the thermodynamic limit. We use the replica method
and extend the procedure of the replica symmetry breaking ansatz to be
applicable in the complex-parameter case. We derive the phase diagrams in the
complex-temperature plane and calculate the density of zeros in each phase.
Near the imaginary axis away from the origin, there is a replica symmetric
phase having a large density. On the other hand, we observe no density in the
spin-glass phases, irrespective of the replica symmetry breaking. We speculate
that this suggests the absence of the temperature chaos. To confirm this, we
investigate the multiple many-body interacting case which is known to exhibit
the chaos effect. The result shows that the density of zeros actually takes
finite values in the spin-glass phase, even on the real axis. These
observations indicate that the density of zeros is more closely connected to
the chaos effect than the replica symmetry breaking.Comment: 22 pages, 8 figure
Cross validation in sparse linear regression with piecewise continuous nonconvex penalties and its acceleration
We investigate the signal reconstruction performance of sparse linear
regression in the presence of noise when piecewise continuous nonconvex
penalties are used. Among such penalties, we focus on the SCAD penalty. The
contributions of this study are three-fold: We first present a theoretical
analysis of a typical reconstruction performance, using the replica method,
under the assumption that each component of the design matrix is given as an
independent and identically distributed (i.i.d.) Gaussian variable. This
clarifies the superiority of the SCAD estimator compared with in a
wide parameter range, although the nonconvex nature of the penalty tends to
lead to solution multiplicity in certain regions. This multiplicity is shown to
be connected to replica symmetry breaking in the spin-glass theory. We also
show that the global minimum of the mean square error between the estimator and
the true signal is located in the replica symmetric phase. Second, we develop
an approximate formula efficiently computing the cross-validation error without
actually conducting the cross-validation, which is also applicable to the
non-i.i.d. design matrices. It is shown that this formula is only applicable to
the unique solution region and tends to be unstable in the multiple solution
region. We implement instability detection procedures, which allows the
approximate formula to stand alone and resultantly enables us to draw phase
diagrams for any specific dataset. Third, we propose an annealing procedure,
called nonconvexity annealing, to obtain the solution path efficiently.
Numerical simulations are conducted on simulated datasets to examine these
results to verify the theoretical results consistency and the approximate
formula efficiency. Another numerical experiment on a real-world dataset is
conducted; its results are consistent with those of earlier studies using the
formulation.Comment: 33 pages, 18 figures. MATLAB codes implementing the proposed method
are distributed in
https://github.com/T-Obuchi/SLRpackage_AcceleratedCV_matla
Accelerating Cross-Validation in Multinomial Logistic Regression with -Regularization
We develop an approximate formula for evaluating a cross-validation estimator
of predictive likelihood for multinomial logistic regression regularized by an
-norm. This allows us to avoid repeated optimizations required for
literally conducting cross-validation; hence, the computational time can be
significantly reduced. The formula is derived through a perturbative approach
employing the largeness of the data size and the model dimensionality. An
extension to the elastic net regularization is also addressed. The usefulness
of the approximate formula is demonstrated on simulated data and the ISOLET
dataset from the UCI machine learning repository.Comment: 30 pages, 9 figures. MATLAB and python codes implementing the formula
derived in the manuscript are distributed in
https://github.com/T-Obuchi/AcceleratedCVonMLR_matlab and
https://github.com/T-Obuchi/AcceleratedCVonMLR_pytho
Weight space structure and analysis using a finite replica number in the Ising perceptron
The weight space of the Ising perceptron in which a set of random patterns is
stored is examined using the generating function of the partition function
as the dimension of the weight vector tends to
infinity, where is the partition function and represents the
configurational average. We utilize for two purposes, depending on
the value of the ratio , where is the number of random
patterns. For , we employ , in
conjunction with Parisi's one-step replica symmetry breaking scheme in the
limit of , to evaluate the complexity that characterizes the number of
disjoint clusters of weights that are compatible with a given set of random
patterns, which indicates that, in typical cases, the weight space is equally
dominated by a single large cluster of exponentially many weights and
exponentially many small clusters of a single weight. For , on the other hand, is used to assess the rate function of a
small probability that a given set of random patterns is atypically separable
by the Ising perceptrons. We show that the analyticity of the rate function
changes at , which implies that the
dominant configuration of the atypically separable patterns exhibits a phase
transition at this critical ratio. Extensive numerical experiments are
conducted to support the theoretical predictions.Comment: 21 pages, 11 figures, Added references, some comments, and
corrections to minor error
Relative species abundance of replicator dynamics with sparse interactions
A theory of relative species abundance on sparsely-connected networks is
presented by investigating the replicator dynamics with symmetric interactions.
Sparseness of a network involves difficulty in analyzing the fixed points of
the equation, and we avoid this problem by treating large self interaction ,
which allows us to construct a perturbative expansion. Based on this
perturbation, we find that the nature of the interactions is directly connected
to the abundance distribution, and some characteristic behaviors, such as
multiple peaks in the abundance distribution and all species coexistence at
moderate values of , are discovered in a wide class of the distribution of
the interactions. The all species coexistence collapses at a critical value of
, , and this collapsing is regarded as a phase transition. To get more
quantitative information, we also construct a non-perturbative theory on random
graphs based on techniques of statistical mechanics. The result shows those
characteristic behaviors are sustained well even for not large . For even
smaller values of , extinct species start to appear and the abundance
distribution becomes rounded and closer to a standard functional form. Another
interesting finding is the non-monotonic behavior of diversity, which
quantifies the number of coexisting species, when changing the ratio of
mutualistic relations . These results are examined by numerical
simulations, and the multiple peaks in the abundance distribution are confirmed
to be robust against a certain level of modifications of the problem. The
numerical results also show that our theory is exact for the case without
extinct species, but becomes less and less precise as the proportion of extinct
species grows.Comment: 27 pages, 14 figure
Replica symmetry breaking, complexity and spin representation in the generalized random energy model
We study the random energy model with a hierarchical structure known as the
generalized random energy model (GREM). In contrast to the original analysis by
the microcanonical ensemble formalism, we investigate the GREM by the canonical
ensemble formalism in conjunction with the replica method. In this analysis,
spin-glass-order parameters are defined for respective hierarchy level, and all
possible patterns of replica symmetry breaking (RSB) are taken into account. As
a result, we find that the higher step RSB ansatz is useful for describing
spin-glass phases in this system. For investigating the nature of the higher
step RSB, we generalize the notion of complexity developed for the one-step RSB
to the higher step and demonstrate how the GREM is characterized by the
generalized complexity. In addition, we propose a novel mean-field spin-glass
model with a hierarchical structure, which is equivalent to the GREM at a
certain limit. We also show that the same hierarchical structure can be
implemented to other mean-field spin models than the GREM. Such models with
hierarchy exhibit phase transitions of multiple steps in common.Comment: 30 pages, 11 figures; minor change
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