Personalized, Sequential, Attentive, Metric-Aware Product Search

The task of personalized product search aims at retrieving a ranked list of products given a user’s input query and his/her purchase history. To address this task, we propose the PSAM model, a Personalized, Sequential, Attentive and Metric-aware (PSAM) model, that learns the semantic representations of three different categories of entities, i.e., users, queries, and products, based on user sequential purchase historical data and the corresponding sequential queries. Specifically, a query-based attentive LSTM (QA-LSTM) model and an attention mechanism are designed to infer users dynamic embeddings, which is able to capture their short-term and long-term preferences. To obtain more fine-grained embeddings of the three categories of entities, a metric-aware objective is deployed in our model to force the inferred embeddings subject to the triangle inequality, which is a more realistic distance measurement for product search. Experiments conducted on four benchmark datasets show that our PSAM model significantly outperforms the state-of-the-art product search baselines in terms of effectiveness by up to 50.9% improvement under NDCG@20. Our visualization experiments further illustrate that the learned product embeddings are able to distinguish different types of products.

A different approach to b(an,bn)-hypermetric spaces

Introduction/purpose: The aim of this paper is to present the concept of b(an,bn)-hypermetric spaces. Methods: Conventional theoretical methods of functional analysis. Results: This study presents the initial results on the topic of b(an,bn)-hypermetric spaces. In the first part, we generalize an n-dimensional (n ≥ 2) hypermetric distance over an arbitrary non-empty set X. The b(an,bn)-hyperdistance function is defined in any way we like, the only constraint being the simultaneous satisfaction of the three properties, viz, non-negativity and positive-definiteness, symmetry and (an, bn)-triangle inequality. In the second part, we discuss the concept of (an, bn)-completeness, with respect to this b(an,bn)-hypermetric, and the fixed point theorem which plays an important role in applied mathematics in a variety of fields. Conclusion: With proper generalisations, it is possible to formulate well-known results of classical metric spaces to the case of b(an,bn)-hypermetric spaces.

A Quantum Geometric Model of Color Similarity Judgements

Since Tversky (1977) argued that similarity judgments violate the three metric axioms, asymmetrical similarity judgments have been offered as particularly difficult challenges for standard, geometric models of similarity, such as multidimensional scaling. According to Tversky (1977), asymmetrical similarity judgments are driven by differences in salience or extent of knowledge. However, the notion of salience has been difficult to operationalize to different kinds of stimuli, especially perceptual stimuli for which there are no apparent differences in extent of knowledge. To investigate similarity judgments between perceptual stimuli, across three experiments we collected data where individuals would rate the similarity of a pair of temporally separated color patches. We identified several violations of symmetry in the empirical results, which the conventional multidimensional scaling model cannot readily capture. Pothos et al. (2013) proposed a quantum geometric model of similarity to account for Tversky’s (1977) findings. In the present work, we developed this model to a form that can be fit to similarity judgments. We fit several variants of quantum and multidimensional scaling models to the behavioral data and concluded in favor of the quantum approach. Without further modifications of the model, the quantum model additionally predicted violations of the triangle inequality that we observed in the same data. Overall, by offering a different form of geometric representation, the quantum geometric model of similarity provides a viable alternative to multidimensional scaling for modeling similarity judgments, while still allowing a convenient, spatial illustration of similarity.

A Social Recommendation Based on Metric Learning and Users’ Co-Occurrence Pattern

For personalized recommender systems, matrix factorization and its variants have become mainstream in collaborative filtering. However, the dot product in matrix factorization does not satisfy the triangle inequality and therefore fails to capture fine-grained information. Metric learning-based models have been shown to be better at capturing fine-grained information than matrix factorization. Nevertheless, most of these models only focus on rating data and social information, which are not sufficient for dealing with the challenges of data sparsity. In this paper, we propose a metric learning-based social recommendation model called SRMC. SRMC exploits users’ co-occurrence patterns to discover their potentially similar or dissimilar users with symmetric relationships and change their relative positions to achieve better recommendations. Experiments on three public datasets show that our model is more effective than the compared models.

On Functions Preserving Products of Certain Classes of Semimetric Spaces

In the paper, we continue the research of Borsík and Doboš on functions which allow us to introduce a metric to the product of metric spaces. We extend their scope to a broader class of spaces which usually fail to satisfy the triangle inequality, albeit they tend to satisfy some weaker form of this axiom. In particular, we examine the behavior of functions preserving b-metric inequality. We provide analogues of the results of Borsík and Doboš adjusted to the new broader setting. The results we obtained are illustrated with multitude of examples. Furthermore, the connections of newly introduced families of functions with the ones already known from the literature are investigated.

A Social Recommendation Based on Metric Learning and Users’ Co-occurrence Pattern

For personalized recommender systems,matrix factorization and its variants have become mainstream in collaborative filtering.However,the dot product in matrix factorization does not satisfy the triangle inequality and therefore fails to capture fine-grained information. Metric learning-based models have been shown to be better at capturing fine-grained information than matrix factorization. Nevertheless,most of these models only focus on rating data and social information, which are not sufficient for dealing with the challenges of data sparsity. In this paper,we propose a metric learning-based social recommendation model called SRMC.SRMC exploits users' co-occurrence pattern to discover their potentially similar or dissimilar users with symmetric relationships and change their relative positions to achieve better recommendations.Experiments on three public datasets show that our model is more effective than the compared models.

Redefining the Graph Edit Distance

Graph edit distance has been used since 1983 to compare objects in machine learning when these objects are represented by attributed graphs instead of vectors. In these cases, the graph edit distance is usually applied to deduce a distance between attributed graphs. This distance is defined as the minimum amount of edit operations (deletion, insertion and substitution of nodes and edges) needed to transform a graph into another. Since now, it has been stated that the distance properties have to be applied [(1) non-negativity (2) symmetry (3) identity and (4) triangle inequality] to the involved edit operations in the process of computing the graph edit distance to make the graph edit distance a metric. In this paper, we show that there is no need to impose the triangle inequality in each edit operation. This is an important finding since in pattern recognition applications, the classification ratio usually maximizes in the edit operation combinations (deletion, insertion and substitution of nodes and edges) that the triangle inequality is not fulfilled.

Exact Acceleration of K-Means++ and K-Means||

K-Means++ and its distributed variant K-Means|| have become de facto tools for selecting the initial seeds of K-means. While alternatives have been developed, the effectiveness, ease of implementation,and theoretical grounding of the K-means++ and || methods have made them difficult to "best" from a holistic perspective. We focus on using triangle inequality based pruning methods to accelerate both of these algorithms to yield comparable or better run-time without sacrificing any of the benefits of these approaches. For both algorithms we are able to reduce distance computations by over 500×. For K-means++ this results in up to a 17×speedup in run-time and a551×speedup for K-means||. We achieve this with simple, but carefully chosen, modifications to known techniques which makes it easy to integrate our approach into existing implementations of these algorithms.

A Greedoid and a Matroid Inspired by Bhargava's $p$-Orderings

Consider a finite set $E$. Assume that each $e \in E$ has a "weight" $w \left(e\right) \in \mathbb{R}$ assigned to it, and any two distinct $e, f \in E$ have a "distance" $d \left(e, f\right) = d \left(f, e\right) \in \mathbb{R}$ assigned to them, such that the distances satisfy the ultrametric triangle inequality $d(a,b)\leqslant \max \left\{d(a,c),d(b,c)\right\}$. We look for a subset of $E$ of given size with maximum perimeter (where the perimeter is defined by summing the weights of all elements and their pairwise distances). We show that any such subset can be found by a greedy algorithm (which starts with the empty set, and then adds new elements one by one, maximizing the perimeter at each step). We use this to define numerical invariants, and also to show that the maximum-perimeter subsets of all sizes form a strong greedoid, and the maximum-perimeter subsets of any given size are the bases of a matroid. This essentially generalizes the "$P$-orderings" constructed by Bhargava in order to define his generalized factorials, and is also similar to the strong greedoid of maximum diversity subsets in phylogenetic trees studied by Moulton, Semple and Steel. We further discuss some numerical invariants of $E, w, d$ stemming from this construction, as well as an analogue where maximum-perimeter subsets are replaced by maximum-perimeter tuples (i.e., elements can appear multiple times).