Computation of Heterogeneous Object Co-embeddings from Relational Measurements

Dimensionality reduction and data embedding methods generate low dimensional representations of a single type of homogeneous data objects. In this work, we examine the problem of generating co-embeddings or pattern representations from two different types of objects within a joint common space of controlled dimensionality, where the only available information is assumed to be a set of pairwise relations or similarities between instances of the two groups. We propose a new method that models the embedding of each object type symmetrically to the other type, subject to flexible scale constraints and weighting parameters. The embedding generation relies on an efficient optimization dispatched using matrix decomposition, that is also extended to support multidimensional co-embeddings. We also propose a scheme of heuristically reducing the parameters of the model, and a simple way of measuring the conformity between the original object relations and the ones re-estimated from the co-embeddings, in order to achieve model selection by identifying the optimal model parameters with a simple search procedure. The capabilities of the proposed method are demonstrated with multiple synthetic and real-world datasets from the text mining domain. The experimental results and comparative analyses indicate that the proposed algorithm outperforms existing methods for co-embedding generation

Evolutionary nonnegative matrix factorization for data compression

This paper aims at improving non-negative matrix factor- ization (NMF) to facilitate data compression. An evolutionary updat- ing strategy is proposed to solve the NMF problem iteratively based on three sets of updating rules including multiplicative, firefly and sur- vival of the fittest rules. For data compression application, the quality of the factorized matrices can be evaluated by measurements such as spar- sity, orthogonality and factorization error to assess compression quality in terms of storage space consumption, redundancy in data matrix and data approximation accuracy. Thus, the fitness score function that drives the evolving procedure is designed as a composite score that takes into account all these measurements. A hybrid initialization scheme is per- formed to improve the rate of convergence, allowing multiple initial can- didates generated by different types of NMF initialization approaches. Effectiveness of the proposed method is demonstrated using Yale and ORL image datasets

Cluster Exploration using Informative Manifold Projections

Dimensionality reduction (DR) is one of the key tools for the visual exploration of high-dimensional data and uncovering its cluster structure in two- or three-dimensional spaces. The vast majority of DR methods in the literature do not take into account any prior knowledge a practitioner may have regarding the dataset under consideration. We propose a novel method to generate informative embeddings which not only factor out the structure associated with different kinds of prior knowledge but also aim to reveal any remaining underlying structure. To achieve this, we employ a linear combination of two objectives: firstly, contrastive PCA that discounts the structure associated with the prior information, and secondly, kurtosis projection pursuit which ensures meaningful data separation in the obtained embeddings. We formulate this task as a manifold optimization problem and validate it empirically across a variety of datasets considering three distinct types of prior knowledge. Lastly, we provide an automated framework to perform iterative visual exploration of high-dimensional data

Quantifying the Informativeness of Similarity Measurements

In this paper, we describe an unsupervised measure for quantifying the 'informativeness' of correlation matrices formed from the pairwise similarities or relationships among data instances. The measure quantifies the heterogeneity of the correlations and is defined as the distance between a correlation matrix and the nearest correlation matrix with constant off-diagonal entries. This non-parametric notion generalizes existing test statistics for equality of correlation coefficients by allowing for alternative distance metrics, such as the Bures and other distances from quantum information theory. For several distance and dissimilarity metrics, we derive closed-form expressions of informativeness, which can be applied as objective functions for machine learning applications. Empirically, we demonstrate that informativeness is a useful criterion for selecting kernel parameters, choosing the dimension for kernel-based nonlinear dimensionality reduction, and identifying structured graphs. We also consider the problem of finding a maximally informative correlation matrix around a target matrix, and explore parameterizing the optimization in terms of the coordinates of the sample or through a lower-dimensional embedding. In the latter case, we find that maximizing the Bures-based informativeness measure, which is maximal for centered rank-1 correlation matrices, is equivalent to minimizing a specific matrix norm, and present an algorithm to solve the minimization problem using the norm's proximal operator. The proposed correlation denoising algorithm consistently improves spectral clustering. Overall, we find informativeness to be a novel and useful criterion for identifying non-trivial correlation structure.

An Interpretable Deep Architecture for Similarity Learning Built Upon Hierarchical Concepts

In general, development of adequately complex mathematical models, such as deep neural networks, can be an effective way to improve the accuracy of learning models. However, this is achieved at the cost of reduced post-hoc model interpretability, because what is learned by the model can become less intelligible and tractable to humans as the model complexity increases. In this paper, we target a similarity learning task in the context of image retrieval, with a focus on the model interpretability issue. An effective similarity neural network (SNN) is proposed not only to seek robust retrieval performance but also to achieve satisfactory post-hoc interpretability. The network is designed by linking the neuron architecture with the organization of a concept tree and by formulating neuron operations to pass similarity information between concepts. Various ways of understanding and visualizing what is learned by the SNN neurons are proposed. We also exhaustively evaluate the proposed approach using a number of relevant datasets against a number of state-of-the-art approaches to demonstrate the effectiveness of the proposed network. Our results show that the proposed approach can offer superior performance when compared against state-of-the-art approaches. Neuron visualization results are demonstrated to support the understanding of the trained neurons