Skein theory and topological quantum field theory

Skein modules arise naturally when mathematicians try to generalize the Jones polynomial of knots. In the first part of this work, we study properties of skein modules. The Temperley-Lieb algebra and some of its generalizations are skein modules. We construct a bases for these skein modules. With this basis, we are able to compute some gram determinants of bilinear forms on these skein modules. Also we use this basis to prove that the Mahler measures of colored Jones polynomial of a sequence of knots converges to the Mahler measure of some two variable polynomial. The topological quantum field theory constructed by Blanchet, Habegger, Mas- baum and Vogel can be considered as a generalization of quantum invariants. It assigns modules to surfaces and linear maps to cobordisms. In particular, it assigns the ground ring to empty surface and constants to cobordisms of empty surface to itself, which are closed 3-manifolds. In this way, we get quantum invariants of 3-manifolds back. In the second part of the work, knot invariants are constructed using topological quantum field theory from quantum invariants of tangles. We prove that this is another way to compute the Turaev-Viro polynomial of knots and related invariants

Approximate sequence alignment

Given a collection of strings and a query string, the goal of the approximate string matching is to efficiently find the strings in the collection, which are similar to the query string. In this paper, we focus on edit distance as a measure to quantify the similarity between two strings. Existing q-gram based methods use inverted lists to index the q-grams of the given string collection. These methods begin with generating the q-grams of the query string, disjoint or overlapping, and then merge the inverted lists of these q-grams. Several filtering techniques have been proposed to segment inverted lists in order to obtain relatively shorter lists, thus reducing the merging cost. The filtering technique we propose in this thesis, which is called position restricted alignment, combines well known length filtering and position filtering to provide more aggressive pruning. We then provide an indexing scheme that integrates the inverted lists storage with the proposed filter. It enables us to auto-filter the inverted lists. We evaluate the effectiveness of the proposed approach by experiments

Efficient XAI Techniques: A Taxonomic Survey

Recently, there has been a growing demand for the deployment of Explainable Artificial Intelligence (XAI) algorithms in real-world applications. However, traditional XAI methods typically suffer from a high computational complexity problem, which discourages the deployment of real-time systems to meet the time-demanding requirements of real-world scenarios. Although many approaches have been proposed to improve the efficiency of XAI methods, a comprehensive understanding of the achievements and challenges is still needed. To this end, in this paper we provide a review of efficient XAI. Specifically, we categorize existing techniques of XAI acceleration into efficient non-amortized and efficient amortized methods. The efficient non-amortized methods focus on data-centric or model-centric acceleration upon each individual instance. In contrast, amortized methods focus on learning a unified distribution of model explanations, following the predictive, generative, or reinforcement frameworks, to rapidly derive multiple model explanations. We also analyze the limitations of an efficient XAI pipeline from the perspectives of the training phase, the deployment phase, and the use scenarios. Finally, we summarize the challenges of deploying XAI acceleration methods to real-world scenarios, overcoming the trade-off between faithfulness and efficiency, and the selection of different acceleration methods.Comment: 15 pages, 3 figure

Simplicial complex representation learning

Simplicial complexes form an important class of topological spaces that are frequently used in many application areas such as computer-aided design, computer graphics, and simulation. Representation learning on graphs, which are just 1-d simplicial complexes, has witnessed a great attention in recent years. However, there has not been enough effort to extend representation learning to higher dimensional simplicial objects due to the additional complexity these objects hold, especially when it comes to entire-simplicial complex representation learning. In this work, we propose a method for simplicial complex-level representation learning that embeds a simplicial complex to a universal embedding space in a way that complex-to-complex proximity is preserved. Our method uses our novel geometric message passing schemes to learn an entire simplicial complex representation in an end-to-end fashion. We demonstrate the proposed model on publicly available mesh dataset. To the best of our knowledge, this work presents the first method for learning simplicial complex-level representation.Comment: MACHINE LEARNING ON GRAPHS, MLoG Workshop at WSDM'2

Approximate String Matching by Position Restricted Alignment ∗

Given a collection of strings, goal of the approximate string matching is to efficiently find the strings in the collection that are similar to a query string. In this paper, we focus on edit distance as measure to quantify the similarity between two strings. Existing q-gram based methods to address this problem use inverted indexes to index the q-grams of given string collection. These methods begin by generating the q-grams of query string (disjoint or overlapping) and then merge the inverted lists of these q-grams. Several filtering techniques have been proposed so as to segment inverted lists to relatively shorter lists thus reducing the merging cost. We use a filtering technique which we call as “position restricted alignment " that combines well known length filtering and position filtering to provide more aggressive pruning. We then provide an indexing scheme that integrates the inverted lists storage with the proposed filter thus enabling us to auto-filter the inverted lists. We evaluate the effectiveness of the proposed approach by thorough experimentation. 1