First-order methods almost always avoid saddle points: the case of
vanishing step-sizes
June 18, 2019
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Mathematics
Optimization and Control
Optimization and Control
In a series of papers \cite{LSJR16, PP17, LPP}, it was established that some
of the most commonly used first order methods almost surely (under random
initializations) and with step-size being small enough, avoid strict saddle
points, as lo...
Phase transition and asymptotic behaviour of flocking Cucker-Smale model
June 18, 2019
Mathematics
Analysis of PDEs
In this paper, we study a continuous ocking Cucker-Smale model with noise,
which has isotropic and polarized stationary solutions depending on the
intensity of the noise. The first result establishes the threshold value of the
noise paramet...
Learning Interpretable Models Using Uncertainty Oracles
June 17, 2019
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Computer Science
Statistics
Machine Learning
Machine Learning
A desirable property of interpretable models is small size, so that they are
easily understandable by humans. This leads to the following challenges: (a)
small sizes typically imply diminished accuracy, and (b) bespoke levers
provided by mo...
Algebraization of Mochizuki's anabelian variation of ring structures,
perfectoid geometry and formal groups
June 17, 2019
Mathematics
Algebraic Geometry
Number Theory
Let M be a multiplicative monoid with identity. Then I show that there is a
universal one dimensional formal group law equipped with an action of M. If
M is p-perfect (i.e. m↦mp is an isomorphism for some prime number
p)...
Abundance of primordial black holes with local non-Gaussianity in peak
theory
June 16, 2019
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Physics
Cosmology and Nongalactic Astrophysics
General Relativity and Quantum Cosmology
We discuss the effect of local type non-Gaussianity on the abundance of
primordial black holes (PBH) based on the peak theory. We provide the PBH
formation criterion based on the so-called compaction function and use the peak
theory statist...
On the discontinuity of the quantum Fisher information for quantum
statistical models with parameter dependent rank
June 14, 2019
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Physics
Quantum Physics
Quantum Physics
We address the discontinuities of the quantum Fisher information (QFI) that
may arise when the parameter of interest takes values that change the rank of
the quantum statistical model. We revisit the classical and the quantum
Cram\'er-Rao t...
The rank of sparse random matrices
June 13, 2019
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Mathematics
Computer Science
Combinatorics
Discrete Mathematics
Probability
We determine the rank of a random matrix over an arbitrary field with
prescribed numbers of non-zero entries in each row and column. As an
application we obtain a formula for the rate of low-density parity check codes.
This formula vindicat...
Stochastic PDEs for large portfolios with general mean-reverting
volatility processes
June 13, 2019
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Mathematics
Quantitative Finance
Probability
Analysis of PDEs
Portfolio Management
Pricing of Securities
Risk Management
We consider a structural stochastic volatility model for the loss from a
large portfolio of credit risky assets. Both the asset value and the volatility
processes are correlated through systemic Brownian motions, with default
determined by ...
Voronoi conjecture for five-dimensional parallelohedra
June 12, 2019
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Mathematics
Combinatorics
Combinatorics
We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely,
we show that if a convex five-dimensional polytope P tiles R5 with
translations, then P is an affine image of the Dirichlet-Voronoi polytope for
a f...
Instant automatic diagnosis of diabetic retinopathy
June 12, 2019
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Electrical Engineering and Systems Science
Computer Science
Image and Video Processing
Computer Vision and Pattern Recognition
The purpose of this study is to evaluate the performance of the OphtAI system
for the automatic detection of referable diabetic retinopathy (DR) and the
automatic assessment of DR severity using color fundus photography. OphtAI
relies on en...