Representations of Affine Quantum Function Algebras

Let $C$ be a symmetrizable generalized Cartan Matrix, and $q$ an indeterminate. {\fg}(C) is the Kac-Moody Lie algebra and U=U_q({\fg}(C)) the associated quantum enveloping algebra over $k={\Bbb Q}(q)$. The quantum function algebra ${\Bbb C}_{q}[G]$ is defined as a suitable $U$-bisubalgebra of the dual space $\hom_{k}(U,k)$ which can be described using matrix elements of integrable $U$-modules. For \fg affine, the highest weight modules of $C_q[G]$ are constructed and, assuming a minimality condition, their (unitarizable) irreducible quotients are shown to be in a 1-1 correspondence with the reduced elements of the Weyl group of ${\frak g}(C)$. Further, these simple module are described in terms of the $C_q[SL_2]$-modules obtained by restriction, and they satisfy a Tensor Product theorem, similar to the finite type case.Comment: 31 pages, adapted from PhD thesis, May 2002, KS

On the optimal estimation of probability measures in weak and strong topologies

Given random samples drawn i.i.d. from a probability measure $\mathbb{P}$ (defined on say, $\mathbb{R}^d$), it is well-known that the empirical estimator is an optimal estimator of $\mathbb{P}$ in weak topology but not even a consistent estimator of its density (if it exists) in the strong topology (induced by the total variation distance). On the other hand, various popular density estimators such as kernel and wavelet density estimators are optimal in the strong topology in the sense of achieving the minimax rate over all estimators for a Sobolev ball of densities. Recently, it has been shown in a series of papers by Gin\'{e} and Nickl that these density estimators on $\mathbb{R}$ that are optimal in strong topology are also optimal in $\|\cdot\|_{\mathcal{F}}$ for certain choices of $\mathcal{F}$ such that $\|\cdot\|_{\mathcal{F}}$ metrizes the weak topology, where $\|\mathbb{P}\|_{\mathcal{F}}:=\sup\{\int f\,\mathrm{d}\mathbb{P}: f\in\mathcal{F}\}$. In this paper, we investigate this problem of optimal estimation in weak and strong topologies by choosing $\mathcal{F}$ to be a unit ball in a reproducing kernel Hilbert space (say $\mathcal{F}_H$ defined over $\mathbb{R}^d$), where this choice is both of theoretical and computational interest. Under some mild conditions on the reproducing kernel, we show that $\|\cdot\|_{\mathcal{F}_H}$ metrizes the weak topology and the kernel density estimator (with $L^1$ optimal bandwidth) estimates $\mathbb{P}$ at dimension independent optimal rate of $n^{-1/2}$ in $\|\cdot\|_{\mathcal{F}_H}$ along with providing a uniform central limit theorem for the kernel density estimator.Comment: Published at http://dx.doi.org/10.3150/15-BEJ713 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A Note on Density Estimation for Binary Sequences

A histogram estimate of the Radon-Nikodym derivative of a probability measure with respect to a dominating measure is developed for binary sequences in $\{0,1\}^{\mathbb{N}}$. A necessary and sufficient condition for the consistency of the estimate in the mean-square sense is given. It is noted that the product topology on $\{0,1\}^{\mathbb{N}}$ and the corresponding dominating product measure pose considerable restrictions on the rate of sampling required for the requisite convergence

On a Clustering Criterion for Dependent Observations

A univariate clustering criterion for stationary processes satisfying a $\beta$-mixing condition is proposed extending the work of \cite{KB2} to the dependent setup. The approach is characterized by an alternative sample criterion function based on truncated partial sums which renders the framework amenable to various interesting extensions for which limit results for partial sums are available. Techniques from empirical process theory for mixing sequences play a vital role in the arguments employed in the proofs of the limit theorems

Two Equivalent Realizations of Trigonometric Dynamical Affine Quantum Group Uq,x(sl2^)=Uq,λ(sl2^)U_{q,x}(\widehat{sl_2})=U_{q,\lambda}(\widehat{sl_2}), Drinfeld Currents and Hopf Algebroid Structures

Two new realizations, denoted $U_{q,x}(\widehat{gl_2})$ and $U(R_{q,x}(\widehat{gl_2}))$ of the trigonometric dynamical quantum affine algebra $U_{q,\lambda}(\widehat{gl_2})$ are proposed, based on Drinfeld-currents and $RLL$ relations respectively, along with a Heisenberg algebra $\left\{P,Q\right\}$, with $x=q^{2P}$. Here $P$ plays the role of the dynamical variable $\lambda$ and $Q=\frac{\partial}{\partial P}$. An explicit isomorphism from $U_{q,x}(\widehat{gl_2})$ to $U(R_{q,x}(\widehat{gl_2}))$ is established, which is a dynamical extension of the Ding-Frenkel isomorphism of $U_{q}(\widehat{gl_2})$ with $U(R_{q}(\widehat{gl_2}))$ between the Drinfeld realization and the Reshetikhin-Tian-Shanksy construction of quantum affine algebras. Hopf algebroid structures and an affine dynamical determinant element are introduced and it is shown that $U_{q,x}(\widehat{sl_2})$ is isomorphic to $U(R_{q,x}(\widehat{sl_2}))$. The dynamical construction is based on the degeneration of the elliptic quantum algebra $U_{q,p}(\widehat{sl_2})$ of Jimbo, Konno et al. as the elliptic variable $p \to 0$.Comment: 37 pages, 2 figures. Misprints corrected and improvements of v

Fast Optimal Bandwidth Selection for RBF Kernel using Reproducing Kernel Hilbert Space Operators for Kernel Based Classifiers

Kernel based methods have shown effective performance in many remote sensing classification tasks. However their performance significantly depend on its hyper-parameters. The conventional technique to estimate the parameter comes with high computational complexity. Thus, the objective of this letter is to propose an fast and efficient method to select the bandwidth parameter of the Gaussian kernel in the kernel based classification methods. The proposed method is developed based on the operators in the reproducing kernel Hilbert space and it is evaluated on Support vector machines and PerTurbo classification method. Experiments conducted with hyperspectral datasets show that our proposed method outperforms the state-of-art method in terms in computational time and classification performance.Comment: Submitted to IEEE GRS

Non-holonomic Constraint Force Postulates

The extended Hamilton's Principle and other methods proposed to handle non-holonomic constraints are considered. They dont agree with each other. By looking at its consistency with D'Alembert's principle for linear non-holonomic constraints, it was claimed in earlier papers that the direct extension of hamilton's principle is incorrect. Nonholonomic Constraints, linear in velocities were considered for this purpose. This paper analyses these claims, and shows that they are incorrect. And hence it shows that it is theoretically impossible to judge any attempt on non-holonomic constraints to be wrong, as long as they are consistent with the D'Alembertian for holonomic constraints.Comment: 9 page

Non-Abelian Geometric Phases Carried by the Spin Fluctuation Tensor

The expectation values of the first and second moments of the quantum mechanical spin operator can be used to define a spin vector and spin fluctuation tensor, respectively. The former is a vector inside the unit ball in three space, while the latter is represented by an ellipsoid in three space. They are both experimentally accessible in many physical systems. By considering transport of the spin vector along loops in the unit ball it is shown that the spin fluctuation tensor picks up geometric phase information. For the physically important case of spin one, the geometric phase is formulated in terms of an SO(3) operator. Loops defined in the unit ball fall into two classes: those which do not pass through the origin and those which pass through the origin. The former class of loops subtend a well defined solid angle at the origin while the latter do not and the corresponding geometric phase is non-Abelian. To deal with both classes, a notion of generalized solid angle is introduced, which helps to clarify the interpretation of the geometric phase information. The experimental systems that can be used to observe this geometric phase are also discussed.Comment: 27 pages, 7 figure

Benford's law: A theoretical explanation for base 2

In this paper, we present a possible theoretical explanation for benford's law. We develop a recursive relation between the probabilities, using simple intuitive ideas. We first use numerical solutions of this recursion and verify that the solutions converge to the benford's law. Finally we solve the recursion analytically to yeild the benford's law for base 2.Comment: 6 page

Few-Shot Learning with Localization in Realistic Settings

Traditional recognition methods typically require large, artificially-balanced training classes, while few-shot learning methods are tested on artificially small ones. In contrast to both extremes, real world recognition problems exhibit heavy-tailed class distributions, with cluttered scenes and a mix of coarse and fine-grained class distinctions. We show that prior methods designed for few-shot learning do not work out of the box in these challenging conditions, based on a new "meta-iNat" benchmark. We introduce three parameter-free improvements: (a) better training procedures based on adapting cross-validation to meta-learning, (b) novel architectures that localize objects using limited bounding box annotations before classification, and (c) simple parameter-free expansions of the feature space based on bilinear pooling. Together, these improvements double the accuracy of state-of-the-art models on meta-iNat while generalizing to prior benchmarks, complex neural architectures, and settings with substantial domain shift.Comment: Appearing in CVPR 2019; added references in covariance pooling sections, added link to code in supplementar