相关论文: VC density and dp rank
Let F be a family of subsets of {1,2,...,n}. The width-degree of an element x in at least one member of F is the width of the family {U in F | x in U}. If F has maximum width-degree at most k, then F is locally k-wide. Bounds on the size of…
The correlation between neutron skin-thickness of a neutron-rich nucleus and slope parameter of symmetry energy is assessed as a function of density using relativistic mean field models containing non-linear couplings among different…
Interpretability of Deep Neural Networks has become a major area of exploration. Although these networks have achieved state of the art accuracy in many tasks, it is extremely difficult to interpret and explain their decisions. In this work…
How many bits of information are required to PAC learn a class of hypotheses of VC dimension $d$? The mathematical setting we follow is that of Bassily et al. (2018), where the value of interest is the mutual information…
The Vapnik-Chervonenkis dimension of a set K in R^n is the maximal dimension of the coordinate cube of a given size, which can be found in coordinate projections of K. We show that the VC dimension of a convex body governs its entropy. This…
Deep convolutional neural networks (CNN) has become the most promising method for object recognition, repeatedly demonstrating record breaking results for image classification and object detection in recent years. However, a very deep CNN…
A collective description of density matrix is presented for identical multi-level atoms, which are either excited initially, driven coherently or pumped incoherently. The density matrix is defined as expectation value of projection or…
We consider the nonparametric estimation of the density function of weakly and strongly dependent processes with noisy observations. We show that in the ordinary smooth case the optimal bandwidth choice can be influenced by long range…
Deep Convolutional Neural Networks (CNNs) have gained great success in image classification and object detection. In these fields, the outputs of all layers of CNNs are usually considered as a high dimensional feature vector extracted from…
We propose a new framework to rank image attractiveness using a novel pairwise deep network trained with a large set of side-by-side multi-labeled image pairs from a web image index. The judges only provide relative ranking between two…
High dimensional data can have a surprising property: pairs of data points may be easily separated from each other, or even from arbitrary subsets, with high probability using just simple linear classifiers. However, this is more of a rule…
Dropout Regularization, serving to reduce variance, is nearly ubiquitous in Deep Learning models. We explore the relationship between the dropout rate and model complexity by training 2,000 neural networks configured with random…
Many systems, including the Internet, social networks, and the power grid, can be represented as graphs. When analyzing graphs, it is often useful to compute scores describing the relative importance or distance between nodes. One example…
In multivariate extreme value analysis, the nature of the extremal dependence between variables should be considered when selecting appropriate statistical models. Interest often lies with determining which subsets of variables can take…
Ecologists have long argued about the strength of density dependence and population regulation, respectively defined as the short-term and long-term rates of return to equilibrium. Here, I give three arguments for the intractability of…
Given a set of points $P\subset \mathbb{R}^{d}$ and a kernel $k$, the Kernel Density Estimate at a point $x\in\mathbb{R}^{d}$ is defined as $\mathrm{KDE}_{P}(x)=\frac{1}{|P|}\sum_{y\in P} k(x,y)$. We study the problem of designing a data…
Bounds on the entropy of patterns of sequences generated by independently identically distributed (i.i.d.) sources are derived. A pattern is a sequence of indices that contains all consecutive integer indices in increasing order of first…
We study the Vapnik-Chervonenkis (VC) density of definable families in certain stable first-order theories. In particular we obtain uniform bounds on VC density of definable families in finite U-rank theories without the finite cover…
We prove that if $E \subseteq \mathbb{R}^d$ ($d\geq 2$) is a Lebesgue-measurable set with density larger than $\frac{n-2}{n-1}$, then $E$ contains similar copies of every $n$-point set $P$ at all sufficiently large scales. Moreover,…
Consider an experiment involving a potentially small number of subjects. Some random variables are observed on each subject: a high-dimensional one called the "observed" random variable, and a one-dimensional one called the "outcome" random…