arXiv subjects

Gargi Ghosh

Publications and source records attributed to Gargi Ghosh.

41 records · Page 3Linked to original sources

arXiv2020

Reducing submodules of Hilbert Modules and Chevalley-Shephard-Todd Theorem

Let $G$ be a finite pseudoreflection group, $Ω\subseteq \mathbb C^n$ be a bounded domain which is a $G$-space and $\mathcal H\subseteq\mathcal O(Ω)$ be an analytic Hilbert module possessing a $G$-invariant reproducing kernel. We study the structure of joint reducing subspaces of the multiplication operator $\mathbf M_{\boldsymbolθ}$ on $\mathcal H,$ where $\{θ_i\}_{i=1}^n$ is a homogeneous system of parameters associated to $G$ and $\boldsymbolθ= (θ_1, \ldots, θ_n)$ is a polynomial map of $\mathbb C^n$. We show that it admits a family $\{\mathbb P_\varrho\mathcal H:\varrho\in\widehat G\}$ of non-trivial joint reducing subspaces, where $\widehat G$ is the set of all equivalence classes of irreducible representations of $G.$ We prove a generalization of Chevalley-Shephard-Todd theorem for the algebra $\mathcal O(Ω)$ of holomorphic functions on $Ω$. As a consequence, we show that for each $\varrho\in \widehat G,$ the multiplication operator $\mathbf M_{\boldsymbolθ}$ on the reducing subspace $\mathbb P_\varrho \mathcal H$ can be realized as multiplication by the coordinate functions on a reproducing kernel Hilbert space of $\mathbb C^{(\mathrm{deg}\,\varrho)^2}$-valued holomorphic functions on $\boldsymbolθ(Ω)$. This, in turn, provides a description of the structure of joint reducing subspaces of the multiplication operator induced by a representative of a proper holomorphic map from a domain $Ω$ in $\mathbb C^n$ which is factored by automorphisms $G\subseteq {\rm Aut}(Ω).$

math.CV↗

arXiv2020

Multiplication operator on the Bergman space by a proper holomorphic map

Suppose that $f := (f_1,\ldots,f_d):Ω_1\toΩ_2$ is a proper holomorphic map between two bounded domains in $\mathbb C^d.$ In this paper, we find a non-trivial minimal joint reducing subspace for the multiplication operator (tuple) $M_f=(M_{f_1},\ldots, M_{f_d})$ on the Bergman space $\mathbb A^2(Ω_1)$, say $\mathcal M.$ We further show that the restriction of $(M_{f_1},\ldots,M_{f_d})$ to $\mathcal M$ is unitarily equivalent to Bergman operator on $\mathbb A^2(Ω_2).$

math.FA↗

arXiv2020

Pre-training via Paraphrasing

We introduce MARGE, a pre-trained sequence-to-sequence model learned with an unsupervised multi-lingual multi-document paraphrasing objective. MARGE provides an alternative to the dominant masked language modeling paradigm, where we self-supervise the reconstruction of target text by retrieving a set of related texts (in many languages) and conditioning on them to maximize the likelihood of generating the original. We show it is possible to jointly learn to do retrieval and reconstruction, given only a random initialization. The objective noisily captures aspects of paraphrase, translation, multi-document summarization, and information retrieval, allowing for strong zero-shot performance on several tasks. For example, with no additional task-specific training we achieve BLEU scores of up to 35.8 for document translation. We further show that fine-tuning gives strong performance on a range of discriminative and generative tasks in many languages, making MARGE the most generally applicable pre-training method to date.

cs.CL↗

arXiv2018

Optimizing Query Evaluations using Reinforcement Learning for Web Search

In web search, typically a candidate generation step selects a small set of documents---from collections containing as many as billions of web pages---that are subsequently ranked and pruned before being presented to the user. In Bing, the candidate generation involves scanning the index using statically designed match plans that prescribe sequences of different match criteria and stopping conditions. In this work, we pose match planning as a reinforcement learning task and observe up to 20% reduction in index blocks accessed, with small or no degradation in the quality of the candidate sets.

cs.IR↗

arXiv2017

Reducing sub-modules of the Bergman module $\mathbb A^{(λ)}(\mathbb D^n)$ under the action of the symmetric group

The weighted Bergman spaces on the polydisc, $\mathbb A^{(λ)}(\mathbb D^n)$, $λ>0,$ splits into orthogonal direct sum of subspaces $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ indexed by the partitions $\boldsymbol p$ of $n,$ which are in one to one correspondence with the equivalence classes of the irreducible representations of the symmetric group on $n$ symbols. In this paper, we prove that each sub-module $\mathbb P_{\boldsymbol p}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ is a locally free Hilbert module of rank equal to square of the dimension $χ_{\boldsymbol p}(1)$ of the corresponding irreducible representation. It is shown that given two partitions $\boldsymbol p$ and $\boldsymbol q$, if $χ_{\boldsymbol p}(1) \ne χ_{\boldsymbol q}(1),$ then the sub-modules $\mathbb P_{\boldsymbol p}\big (\mathbb A^{(λ)}(\mathbb D^n)\big )$ and $\mathbb P_{\boldsymbol q}\big (\mathbb A^{(λ)}(\mathbb D^n)\big )$ are not equivalent. We prove that for the trivial and the sign representation corresponding to the partitions $\boldsymbol p = (n)$ and $\boldsymbol p = (1,\ldots,1)$, respectively, the sub-modules $\mathbb P_{(n)}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ and $\mathbb P_{(1,\ldots,1)}\big(\mathbb A^{(λ)}(\mathbb D^n)\big)$ are inequivalent. In particular, for $n=3$, we show that all the sub-modules in this decomposition are inequivalent.

math.FA↗