1,563 research outputs found

### Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

In this paper, we establish that for a wide class of controlled stochastic
differential equations (SDEs) with stiff coefficients, the value functions of
corresponding zero-sum games can be represented by a deep artificial neural
network (DNN), whose complexity grows at most polynomially in both the
dimension of the state equation and the reciprocal of the required accuracy.
Such nonlinear stiff systems may arise, for example, from Galerkin
approximations of controlled stochastic partial differential equations (SPDEs),
or controlled PDEs with uncertain initial conditions and source terms. This
implies that DNNs can break the curse of dimensionality in numerical
approximations and optimal control of PDEs and SPDEs. The main ingredient of
our proof is to construct a suitable discrete-time system to effectively
approximate the evolution of the underlying stochastic dynamics. Similar ideas
can also be applied to obtain expression rates of DNNs for value functions
induced by stiff systems with regime switching coefficients and driven by
general L\'{e}vy noise.Comment: This revised version has been accepted for publication in Analysis
and Application

### Weighted-Sampling Audio Adversarial Example Attack

Recent studies have highlighted audio adversarial examples as a ubiquitous
threat to state-of-the-art automatic speech recognition systems. Thorough
studies on how to effectively generate adversarial examples are essential to
prevent potential attacks. Despite many research on this, the efficiency and
the robustness of existing works are not yet satisfactory. In this paper, we
propose~\textit{weighted-sampling audio adversarial examples}, focusing on the
numbers and the weights of distortion to reinforce the attack. Further, we
apply a denoising method in the loss function to make the adversarial attack
more imperceptible. Experiments show that our method is the first in the field
to generate audio adversarial examples with low noise and high audio robustness
at the minute time-consuming level.Comment: https://aaai.org/Papers/AAAI/2020GB/AAAI-LiuXL.9260.pd

### Totally non-negativity of a family of change-of-basis matrices

Let ${\bf a}=(a_1, a_2, \ldots, a_n)$ and ${\bf e}=(e_1, e_2, \ldots, e_n)$
be real sequences. Denote by $M_{{\bf e}\rightarrow {\bf a}}$ the
$(n+1)\times(n+1)$ matrix whose $(m,k)$ entry ($m, k \in \{0,\ldots, n\}$) is
the coefficient of the polynomial $(x-a_1)\cdots(x-a_k)$ in the expansion of
$(x-e_1)\cdots(x-e_m)$ as a linear combination of the polynomials $1, x-a_1,
\ldots, (x-a_1)\cdots(x-a_m)$. By appropriate choice of ${\bf a}$ and ${\bf e}$
the matrix $M_{{\bf e}\rightarrow {\bf a}}$ can encode many familiar
doubly-indexed combinatorial sequences, such as binomial coefficients, Stirling
numbers of both kinds, Lah numbers and central factorial numbers.
In all four of these examples, $M_{{\bf e}\rightarrow {\bf a}}$ enjoys the
property of total non-negativity -- the determinants of all its square
submatrices are non-negative. This leads to a natural question: when, in
general, is $M_{{\bf e}\rightarrow {\bf a}}$ totally non-negative?
Galvin and Pacurar found a simple condition on ${\bf e}$ that characterizes
total non-negativity of $M_{{\bf e}\rightarrow {\bf a}}$ when ${\bf a}$ is
non-decreasing. Here we fully extend this result. For arbitrary real sequences
${\bf a}$ and ${\bf e}$, we give a condition that can be checked in $O(n^2)$
time that determines whether $M_{{\bf e}\rightarrow {\bf a}}$ is totally
non-negative. When $M_{{\bf e}\rightarrow {\bf a}}$ is totally non-negative, we
witness this with a planar network whose weights are non-negative and whose
path matrix is $M_{{\bf e}\rightarrow {\bf a}}$. When it is not, we witness
this with an explicit negative minor.Comment: Some small errors from earlier version have been correcte

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