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## Linear Convergence of Stochastic Iterative Greedy Algorithms with Sparse Constraints

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/06/2014

Relevância na Pesquisa

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#Mathematics - Numerical Analysis#Computer Science - Information Theory#Mathematics - Optimization and Control#41A46, 68Q25, 68W20, 90C27, 65B99, 52A99, 60G99, 62L20

Motivated by recent work on stochastic gradient descent methods, we develop
two stochastic variants of greedy algorithms for possibly non-convex
optimization problems with sparsity constraints. We prove linear convergence in
expectation to the solution within a specified tolerance. This generalized
framework applies to problems such as sparse signal recovery in compressed
sensing, low-rank matrix recovery, and covariance matrix estimation, giving
methods with provable convergence guarantees that often outperform their
deterministic counterparts. We also analyze the settings where gradients and
projections can only be computed approximately, and prove the methods are
robust to these approximations. We include many numerical experiments which
align with the theoretical analysis and demonstrate these improvements in
several different settings.

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## Estimation of a k-monotone density, part 1: characterizations consistency, and minimax lower bounds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/09/2005

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Shape constrained densities are encountered in many nonparametric estimation
problems. The classes of monotone or convex (and monotone) densities can be
viewed as special cases of the classes of k-monotone densities. A density g is
said to be k-monotone if its successive derivatives up to order k-2 are
alternately negative or positive, with the k-2 derivative convex (or concave)
according as k is even (or odd) respectively. g is simply nonincreasing if k=1.
These classes of shape constrained densities bridge the gap between the classes
of monotone (1-monotone) and convex decreasing (2-monotone) densities for which
asymptotic results are known, and the class of completely monotone densities
with k infinite.
In this series of four papers we consider both (nonparametric) Maximum
Likelihood estimators and Least Squares estimators of a k-monotone density. In
this first part (part 1), we prove existence of the estimators and give
characterizations. We also establish consistency properties, and show that the
estimators are splines of order k (degree k-1) with simple knots. We further
provide asymptotic minimax risk lower bounds for estimating a k-monotone
density g_0 (x_0) and its derivatives g_0^{(j)}(x_0), for j=1, ..., k-1, at a
fixed point x_0 under the assumption that the k-th derivative of g_0 at x_0 is
non-zero.; Comment: 60 pages. Submitted to Annals of Statistics. See also
http://www.stat.washington.edu/www/research/reports/2004/tr459.pdf
http://www.stat.washington.edu/jaw/RESEARCH/PAPERS/available.html

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## Estimation of a $k$-monotone density: limit distribution theory and the spline connection

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

16.59%

We study the asymptotic behavior of the Maximum Likelihood and Least Squares
Estimators of a $k$-monotone density $g_0$ at a fixed point $x_0$ when $k>2$.
We find that the $j$th derivative of the estimators at $x_0$ converges at the
rate $n^{-(k-j)/(2k+1)}$ for $j=0,...,k-1$. The limiting distribution depends
on an almost surely uniquely defined stochastic process $H_k$ that stays above
(below) the $k$-fold integral of Brownian motion plus a deterministic drift
when $k$ is even (odd). Both the MLE and LSE are known to be splines of degree
$k-1$ with simple knots. Establishing the order of the random gap
$\tau_n^+-\tau_n^-$, where $\tau_n^{\pm}$ denote two successive knots, is a key
ingredient of the proof of the main results. We show that this ``gap problem''
can be solved if a conjecture about the upper bound on the error in a
particular Hermite interpolation via odd-degree splines holds.; Comment: Published in at http://dx.doi.org/10.1214/009053607000000262 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org)

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## Catastrophe Management and Inter-Reserve Distance for Marine Reserve Networks

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematics - Optimization and Control#Mathematics - Probability#Quantitative Biology - Other Quantitative Biology#Quantitative Biology - Populations and Evolution#Quantitative Biology - Quantitative Methods#60G99, 60J10, 60J25, 90B15, 92B05

We consider the optimal spacing between marine reserves for maximising the
viability of a species occupying a reserve network. The closer the networks are
placed together, the higher the probability of colonisation of an empty reserve
by an occupied reserve, thus increasing population viability. However, the
closer the networks are placed together, the higher the probability that a
catastrophe will cause extinction of the species in both reserves, thus
decreasing population viability. Using a simple discrete-time Markov chain
model for the presence or absence of the species in each reserve we determine
the distance between the two reserves which provides the optimal trade-off
between these processes, resulting in maximum viability of the species.; Comment: 12 pages and 9 figures

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## Utility Maximization with a Stochastic Clock and an Unbounded Random Endowment

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/03/2005

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#Mathematics - Probability#Quantitative Finance - Computational Finance#91B28 (Primary) 60G99, 60H99. (Secondary)

We introduce a linear space of finitely additive measures to treat the
problem of optimal expected utility from consumption under a stochastic clock
and an unbounded random endowment process. In this way we establish existence
and uniqueness for a large class of utility-maximization problems including the
classical ones of terminal wealth or consumption, as well as the problems that
depend on a random time horizon or multiple consumption instances. As an
example we explicitly treat the problem of maximizing the logarithmic utility
of a consumption stream, where the local time of an Ornstein-Uhlenbeck process
acts as a stochastic clock.; Comment: Published at http://dx.doi.org/10.1214/105051604000000738 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org)

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## Intersection probabilities for a chordal SLE path and a semicircle

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We derive a number of estimates for the probability that a chordal SLE path
in the upper half plane H intersects a semicircle centred on the real line. We
prove that if 00 of radius rx, 00. For 4

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## Free-Knot Spline Approximation of Stochastic Processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/12/2006

Relevância na Pesquisa

16.59%

We study optimal approximation of stochastic processes by polynomial splines
with free knots. The number of free knots is either a priori fixed or may
depend on the particular trajectory. For the $s$-fold integrated Wiener process
as well as for scalar diffusion processes we determine the asymptotic behavior
of the average $L_p$-distance to the splines spaces, as the (expected) number
$k$ of free knots tends to infinity.; Comment: 23 pages

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## Expected Number of Slope Crossings of Certain Gaussian Random Polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/12/2006

Relevância na Pesquisa

16.59%

Let $Q_n(x)=\sum_{i=0}^{n} A_{i}x^{i}$ be a random polynomial where the
coefficients
$A_0,A_1,... $ form a sequence of centered Gaussian random variables.
Moreover, assume that the increments $\Delta_j=A_j-A_{j-1}$, $j=0,1,2,...$ are
independent, assuming $A_{-1}=0$. The coefficients can be considered as $n$
consecutive observations of a Brownian motion. We study the number of times
that such a random polynomial crosses a line which is not necessarily parallel
to the x-axis. More precisely we obtain the asymptotic behavior of the expected
number of real roots of the equation $Q_n(x)=Kx$, for the cases that $K$ is any
non-zero real constant $K=o(n^{1/4})$, and $K=o(n^{1/2})$ separately.; Comment: 11 pages

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## Limit theorems for stochastic approximation algorithms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/02/2011

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16.59%

We prove a central limit theorem applicable to one dimensional stochastic
approximation algorithms that converge to a point where the error terms of the
algorithm do not vanish. We show how this applies to a certain class of these
algorithms that in particular covers a generalized P\'olya urn model, which is
also discussed. In addition, we show how to scale these algorithms in some
cases where we cannot determine the limiting distribution but expect it to be
non-normal.

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## The Rademacher cotype of operators from $l_\infty^N$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

16.59%

We show that for any operator $T:l_\infty^N\to Y$, where $Y$ is a Banach
space, that its cotype 2 constant, $K_2(T)$, is related to its $(2,1)$-summing
norm, $\pi_{2,1}(T)$, by $K_2(T) \le c \log\log N \pi_{2,1}(T) $. Thus, we can
show that there is an operator $T:C(K)\to Y$ that has cotype 2, but is not
2-summing.

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## Asymptotic estimates on the von Neumann inequality for homogeneous polynomials

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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#Mathematics - Functional Analysis#Mathematics - Operator Algebras#47A13, 47A60, 28A78, 60G99, 46G25, 05B05

By the von Neumann inequality for homogeneous polynomials there exists a
positive constant $C_{k,q}(n)$ such that for every $k$-homogeneous polynomial
$p$ in $n$ variables and every $n$-tuple of commuting operators $(T_1, \dots,
T_n)$ with $\sum_{i=1}^{n} \Vert T_{i} \Vert^{q} \leq 1$ we have \[ \|p(T_1,
\dots, T_n)\|_{\mathcal L(\mathcal H)} \leq C_{k,q}(n) \; \sup\{ |p(z_1, \dots,
z_n)| : \textstyle \sum_{i=1}^{n} \vert z_{i} \vert^{q} \leq 1 \}\,. \] For
fixed $k$ and $q$, we study the asymptotic growth of the smallest constant
$C_{k,q}(n)$ as $n$ (the number of variables/operators) tends to infinity. For
$q = \infty$, we obtain the correct asymptotic behavior of this constant
(answering a question posed by Dixon in the seventies). For $2 \leq q < \infty$
we improve some lower bounds given by Mantero and Tonge, and prove the
asymptotic behavior up to a logarithmic factor. To achieve this we provide
estimates of the norm of homogeneous unimodular Steiner polynomials, i.e.
polynomials such that the multi-indices corresponding to the nonzero
coefficients form partial Steiner systems.; Comment: 14 pages, Accepted in Journal f\"ur die reine und angewandte
Mathematik (Crelle's Journal)

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## Logarithmic Coefficients and Multifractality of Whole-Plane SLE

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/04/2015

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#Mathematical Physics#Condensed Matter - Statistical Mechanics#Mathematics - Complex Variables#Mathematics - Probability#28A80, 30C50, 30C99, 35K10, 60G51, 60G99, 60J67

We consider the whole-plane SLE conformal map f from the unit disk to the
slit plane, and show that its mixed moments, involving a power p of the
derivative modulus |f'| and a power q of the map |f| itself, have closed forms
along some integrability curves in the (p,q) moment plane, which depend
continuously on the SLE parameter kappa. The generalization of this
integrability property to the m-fold transform of f is also given. We define a
generalized integral means spectrum corresponding to the singular behavior of
the mixed moments above. The average generalized spectrum of whole-plane SLE
takes four possible forms, separated by five phase transition lines in the
moment plane, whereas the average generalized spectrum of the m-fold
whole-plane SLE is directly obtained from a linear map acting in that plane. We
also conjecture the form of the universal generalized integral means spectrum.; Comment: 44 pages, 17 figures

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## Stochastic Gradient Descent, Weighted Sampling, and the Randomized Kaczmarz algorithm

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

16.59%

#Mathematics - Numerical Analysis#Computer Science - Computer Vision and Pattern Recognition#Computer Science - Learning#Mathematics - Optimization and Control#Statistics - Machine Learning#65B99, 52A99, 60G99, 62L20

We obtain an improved finite-sample guarantee on the linear convergence of
stochastic gradient descent for smooth and strongly convex objectives,
improving from a quadratic dependence on the conditioning $(L/\mu)^2$ (where
$L$ is a bound on the smoothness and $\mu$ on the strong convexity) to a linear
dependence on $L/\mu$. Furthermore, we show how reweighting the sampling
distribution (i.e. importance sampling) is necessary in order to further
improve convergence, and obtain a linear dependence in the average smoothness,
dominating previous results. We also discuss importance sampling for SGD more
broadly and show how it can improve convergence also in other scenarios. Our
results are based on a connection we make between SGD and the randomized
Kaczmarz algorithm, which allows us to transfer ideas between the separate
bodies of literature studying each of the two methods. In particular, we recast
the randomized Kaczmarz algorithm as an instance of SGD, and apply our results
to prove its exponential convergence, but to the solution of a weighted least
squares problem rather than the original least squares problem. We then present
a modified Kaczmarz algorithm with partially biased sampling which does
converge to the original least squares solution with the same exponential
convergence rate.; Comment: 22 pages...

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## Reflection positive stochastic processes indexed by Lie groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/10/2015

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In physics reflection positivity is a bridge between euclidean quantum theory
and quantum field theory. In mathematics it is Cartan duality of symmetric Lie
groups and unitary representations. In this discussion causality is represented
by an involutive semigroup in $G$. We connect those ideas to stochastic
processes indexed by a Lie group emphasizing Markov processes and measures on
path spaces for topological groups. We also explain when Gaussian measures can
be realized in the space of distribution vectors of a unitary representation.; Comment: 47 pages

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## Measure changes with extinction

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We consider a change of measure by a martingale $Z_t$ and clarify that in
general $1/Z_t$ is only a supermartingale under the changed measure. We then
give a necessary and sufficient condition for the event that the limit of the
martingale is zero to coincide with the event that the martingale hits zero in
finite time (up to a set of zero probability).; Comment: 6 pages; corrected typo, shortened proof of Theorem 6

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## Convergence in Density in Finite Time Windows and the Skorohod Representation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/08/2015

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According to the Dudley-Wichura extension of the Skorohod representation
theorem, convergence in distribution to a limit in a separable set is
equivalent to the existence of a coupling with elements converging a.s. in the
metric. A density analogue of this theorem says that a sequence of probability
densities on a general measurable space has a probability density as a
pointwise lower limit if and only if there exists a coupling with elements
converging a.s. in the discrete metric. In this paper the discrete-metric
theorem is extended to stochastic processes considered in a widening time
window. The extension is then used to prove the separability version of the
Skorohod representation theorem.

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## On the Optimal Amount of Experimentation in Sequential Decision Problems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/07/2009

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We provide a tight bound on the amount of experimentation under the optimal
strategy in sequential decision problems. We show the applicability of the
result by providing a bound on the cut-off in a one-arm bandit problem.

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## Chains with unbounded variable length memory: perfect simulation and visible regeneration scheme

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We present a new perfect simulation algorithm for stationary chains having
unbounded variable length memory. This is the class of infnite memory chains
for which the family of transition probabilities is represented by a
probabilistic context tree. We do not assume any continuity condition: our
condition is expressed in terms of the structure of the context tree. More
precisely, the length of the contexts is a deterministic function of the
distance to the last occurrence of some determined string of symbols. It turns
out that the resulting class of chains can be seen as a natural extension of
the class of chains having a renewal string. In particular, our chains exhibit
a visible regeneration scheme.; Comment: 27 pages, 10 figures, slight improvements of the results and
simplification of the proof, simulations.

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## A Distributed Procedure for Computing Stochastic Expansions with Mathematica

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/09/2010

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The solution of a (stochastic) differential equation can be locally
approximated by a (stochastic) expansion. If the vector field of the
differential equation is a polynomial, the corresponding expansion is a linear
combination of iterated integrals of the drivers and can be calculated using
Picard Iterations. However, such expansions grow exponentially fast in their
number of terms, due to their specific algebra, rendering their practical use
limited.
We present a Mathematica procedure that addresses this issue by
re-parametrising the polynomials and distributing the load in as small as
possible parts that can be processed and manipulated independently, thus
alleviating large memory requirements and being perfectly suited for
parallelized computation. We also present an iterative implementation of the
shuffle product (as opposed to a recursive one, more usually implemented) as
well as a fast way for calculating the expectation of iterated Stratonovich
integrals for Brownian Motion.; Comment: 15 pages, 2 figures. Submitted

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## A continuous time stochastic model for biological neural nets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/07/2015

Relevância na Pesquisa

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We propose a new stochastic model for biological neural nets which is a
continuous time version of the model proposed by Galves and L\"ocherbach in [A.
Galves and E. L\"ocherbach, "Infinite systems of interacting chains with
memory of variable length - a stochastic model for biological neural nets", J.
Stat. Phys. 151 (2013), no. 5, 896-921]. We also show how to computationally
simulate such model for easy neuron potential decays and probability functions
and characterize when the model has a finite time of death almost surely.; Comment: 17 pages

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