Mauricio Flores

Luisa F. Polania, Mauricio Flores, Yiran Li et al.

We propose a graph neural network (GNN) approach to the problem of predicting the stylistic compatibility of a set of furniture items from images. While most existing results are based on siamese networks which evaluate pairwise compatibility between items, the proposed GNN architecture exploits relational information among groups of items. We present two GNN models, both of which comprise a deep CNN that extracts a feature representation for each image, a gated recurrent unit (GRU) network that models interactions between the furniture items in a set, and an aggregation function that calculates the compatibility score. In the first model, a generalized contrastive loss function that promotes the generation of clustered embeddings for items belonging to the same furniture set is introduced. Also, in the first model, the edge function between nodes in the GRU and the aggregation function are fixed in order to limit model complexity and allow training on smaller datasets; in the second model, the edge function and aggregation function are learned directly from the data. We demonstrate state-of-the art accuracy for compatibility prediction and "fill in the blank" tasks on the Bonn and Singapore furniture datasets. We further introduce a new dataset, called the Target Furniture Collections dataset, which contains over 6000 furniture items that have been hand-curated by stylists to make up 1632 compatible sets. We also demonstrate superior prediction accuracy on this dataset.

Mauricio Flores, Jeff Calder, Gilad Lerman

This paper addresses theory and applications of $\ell_p$-based Laplacian regularization in semi-supervised learning. The graph $p$-Laplacian for $p>2$ has been proposed recently as a replacement for the standard ($p=2$) graph Laplacian in semi-supervised learning problems with very few labels, where Laplacian learning is degenerate. In the first part of the paper we prove new discrete to continuum convergence results for $p$-Laplace problems on $k$-nearest neighbor ($k$-NN) graphs, which are more commonly used in practice than random geometric graphs. Our analysis shows that, on $k$-NN graphs, the $p$-Laplacian retains information about the data distribution as $p\to \infty$ and Lipschitz learning ($p=\infty$) is sensitive to the data distribution. This situation can be contrasted with random geometric graphs, where the $p$-Laplacian forgets the data distribution as $p\to \infty$. We also present a general framework for proving discrete to continuum convergence results in graph-based learning that only requires pointwise consistency and monotonicity. In the second part of the paper, we develop fast algorithms for solving the variational and game-theoretic $p$-Laplace equations on weighted graphs for $p>2$. We present several efficient and scalable algorithms for both formulations, and present numerical results on synthetic data indicating their convergence properties. Finally, we conduct extensive numerical experiments on the MNIST, FashionMNIST and EMNIST datasets that illustrate the effectiveness of the $p$-Laplacian formulation for semi-supervised learning with few labels. In particular, we find that Lipschitz learning ($p=\infty$) performs well with very few labels on $k$-NN graphs, which experimentally validates our theoretical findings that Lipschitz learning retains information about the data distribution (the unlabeled data) on $k$-NN graphs.