29,711 research outputs found
Loss minimization yields multicalibration for large neural networks
Multicalibration is a notion of fairness that aims to provide accurate
predictions across a large set of groups. Multicalibration is known to be a
different goal than loss minimization, even for simple predictors such as
linear functions. In this note, we show that for (almost all) large neural
network sizes, optimally minimizing squared error leads to multicalibration.
Our results are about representational aspects of neural networks, and not
about algorithmic or sample complexity considerations. Previous such results
were known only for predictors that were nearly Bayes-optimal and were
therefore representation independent. We emphasize that our results do not
apply to specific algorithms for optimizing neural networks, such as SGD, and
they should not be interpreted as "fairness comes for free from optimizing
neural networks"
Resource-efficient high-dimensional entanglement detection via symmetric projections
We introduce two families of criteria for detecting and quantifying the
entanglement of a bipartite quantum state of arbitrary local dimension. The
first is based on measurements in mutually unbiased bases and the second is
based on equiangular measurements. Both criteria give a qualitative result in
terms of the state's entanglement dimension and a quantitative result in terms
of its fidelity with the maximally entangled state. The criteria are
universally applicable since no assumptions on the state are required.
Moreover, the experimenter can control the trade-off between
resource-efficiency and noise-tolerance by selecting the number of measurements
performed. For paradigmatic noise models, we show that only a small number of
measurements are necessary to achieve nearly-optimal detection in any
dimension. The number of global product projections scales only linearly in the
local dimension, thus paving the way for detection and quantification of very
high-dimensional entanglement.Comment: 6+2 page
Classical model emerges in quantum entanglement: Quantum Monte Carlo study for an Ising-Heisenberg bilayer
By developing a cluster sampling of stochastic series expansion quantum Monte
Carlo method, we investigate a spin- model on a bilayer square lattice
with intra-layer ferromagnetic (FM) Ising coupling and inter-layer
antiferromagnetic Heisenberg interaction. The continuous quantum phase
transition which occurs at between the FM Ising phase and the
dimerized phase is studied via large scale simulations. From the analyzes of
critical exponents we show that this phase transition belongs to the
(2+1)-dimensional Ising universality class. Besides, the quantum entanglement
is strong between the two layers, especially in dimerized phase. The effective
Hamiltonian of single layer seems like a transverse field Ising model. However,
we found the quantum entanglement Hamiltonian is a pure classical Ising model
without any quantum fluctuations. Furthermore, we give a more general
explanation about how a classical entanglement Hamiltonian emerges
Positive Geometries of S-matrix without Color
In this note, we prove that the realization of associahedron discovered by
Arkani-Hamed, Bai, He, and Yun (ABHY) is a positive geometry for tree-level
S-matrix of scalars which have no color and which interact via cubic coupling.
More in detail, we consider diffeomorphic images of the ABHY associahedron. The
diffeomorphisms are linear maps parametrized by the right cosets of the
Dihedral group on n elements. The set of all the boundaries associated with
these copies of ABHY associahedron exhaust all the simple poles. We prove that
the sum over the diffeomorphic copies of ABHY associahedron is a positive
geometry and the total volume obtained by summing over all the dual
associahedra is proportional to the tree-level S matrix of (massive or
massless) scalar particles with cubic coupling. We then provide non-trivial
evidence that the projection of the planar scattering forms parametrized by the
Stokes polytope on these realizations of the associahedron leads to the
tree-level amplitudes of scalar particles, which interact via quartic coupling.
Our results build on ideas laid out in our previous works, leading to further
evidence that a large class of positive geometries which are diffeomorphic to
the ABHY associahedron defines an ``amplituhedron" for a tree-level S matrix of
some local and unitary scalar theory. We also highlight a fundamental
obstruction in applying these ideas to discover positive geometry for the one
loop integrand when propagating states have no color.Comment: 33 Pages, 4 Figure
Soliton Gas: Theory, Numerics and Experiments
The concept of soliton gas was introduced in 1971 by V. Zakharov as an
infinite collection of weakly interacting solitons in the framework of
Korteweg-de Vries (KdV) equation. In this theoretical construction of a diluted
soliton gas, solitons with random parameters are almost non-overlapping. More
recently, the concept has been extended to dense gases in which solitons
strongly and continuously interact. The notion of soliton gas is inherently
associated with integrable wave systems described by nonlinear partial
differential equations like the KdV equation or the one-dimensional nonlinear
Schr\"odinger equation that can be solved using the inverse scattering
transform. Over the last few years, the field of soliton gases has received a
rapidly growing interest from both the theoretical and experimental points of
view. In particular, it has been realized that the soliton gas dynamics
underlies some fundamental nonlinear wave phenomena such as spontaneous
modulation instability and the formation of rogue waves. The recently
discovered deep connections of soliton gas theory with generalized
hydrodynamics have broadened the field and opened new fundamental questions
related to the soliton gas statistics and thermodynamics. We review the main
recent theoretical and experimental results in the field of soliton gas. The
key conceptual tools of the field, such as the inverse scattering transform,
the thermodynamic limit of finite-gap potentials and the Generalized Gibbs
Ensembles are introduced and various open questions and future challenges are
discussed.Comment: 35 pages, 8 figure
Jack Derangements
For each integer partition we give a simple combinatorial
expression for the sum of the Jack character over the
integer partitions of with no singleton parts. For this
gives closed forms for the eigenvalues of the permutation and perfect matching
derangement graphs, resolving an open question in algebraic graph theory. A
byproduct of the latter is a simple combinatorial formula for the immanants of
the matrix where is the all-ones matrix, which might be of
independent interest. Our proofs center around a Jack analogue of a hook
product related to Cayley's --process in classical invariant theory,
which we call the principal lower hook product
Experimental Evidence of Accelerated Seismic Release without Critical Failure in Acoustic Emissions of Compressed Nanoporous Materials
The total energy of acoustic emission (AE) events in externally stressed materials diverges when approaching macroscopic failure. Numerical and conceptual models explain this accelerated seismic release (ASR) as the approach to a critical point that coincides with ultimate failure. Here, we report ASR during soft uniaxial compression of three silica-based ( SiO2) nanoporous materials. Instead of a singular critical point, the distribution of AE energies is stationary, and variations in the activity rate are sufficient to explain the presence of multiple periods of ASR leading to distinct brittle failure events. We propose that critical failure is suppressed in the AE statistics by mechanisms of transient hardening. Some of the critical exponents estimated from the experiments are compatible with mean field models, while others are still open to interpretation in terms of the solution of frictional and fracture avalanche models
Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight
We consider 2D flows with a homoclinic figure-eight to a dissipative saddle. We study the rich dynamics that such a system exhibits under a periodic forcing. First, we derive the bifurcation diagram using topological techniques. In particular, there is a homoclinic zone in the parameter space with a non-smooth boundary. We provide a complete explanation of this phenomenon relating it to primary quadratic homoclinic tangency curves which end up at some cubic tangency (cusp) points. We also describe the possible attractors that exist (and may coexist) in the system. A main goal of this work is to show how the previous qualitative description can be complemented with quantitative global information. To this end, we introduce a return map model which can be seen as the simplest one which is 'universal' in some sense. We carry out several numerical experiments on the model, to check that all the objects predicted to exist by the theory are found in the model, and also to investigate new properties of the system
Nonparametric Two-Sample Test for Networks Using Joint Graphon Estimation
This paper focuses on the comparison of networks on the basis of statistical
inference. For that purpose, we rely on smooth graphon models as a
nonparametric modeling strategy that is able to capture complex structural
patterns. The graphon itself can be viewed more broadly as density or intensity
function on networks, making the model a natural choice for comparison
purposes. Extending graphon estimation towards modeling multiple networks
simultaneously consequently provides substantial information about the
(dis-)similarity between networks. Fitting such a joint model - which can be
accomplished by applying an EM-type algorithm - provides a joint graphon
estimate plus a corresponding prediction of the node positions for each
network. In particular, it entails a generalized network alignment, where
nearby nodes play similar structural roles in their respective domains. Given
that, we construct a chi-squared test on equivalence of network structures.
Simulation studies and real-world examples support the applicability of our
network comparison strategy.Comment: 25 pages, 6 figure
When to be critical? Performance and evolvability in different regimes of neural Ising agents
It has long been hypothesized that operating close to the critical state is
beneficial for natural, artificial and their evolutionary systems. We put this
hypothesis to test in a system of evolving foraging agents controlled by neural
networks that can adapt agents' dynamical regime throughout evolution.
Surprisingly, we find that all populations that discover solutions, evolve to
be subcritical. By a resilience analysis, we find that there are still benefits
of starting the evolution in the critical regime. Namely, initially critical
agents maintain their fitness level under environmental changes (for example,
in the lifespan) and degrade gracefully when their genome is perturbed. At the
same time, initially subcritical agents, even when evolved to the same fitness,
are often inadequate to withstand the changes in the lifespan and degrade
catastrophically with genetic perturbations. Furthermore, we find the optimal
distance to criticality depends on the task complexity. To test it we introduce
a hard and simple task: for the hard task, agents evolve closer to criticality
whereas more subcritical solutions are found for the simple task. We verify
that our results are independent of the selected evolutionary mechanisms by
testing them on two principally different approaches: a genetic algorithm and
an evolutionary strategy. In summary, our study suggests that although optimal
behaviour in the simple task is obtained in a subcritical regime, initializing
near criticality is important to be efficient at finding optimal solutions for
new tasks of unknown complexity.Comment: arXiv admin note: substantial text overlap with arXiv:2103.1218
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