Francisco Escolano

Commute Times in Dense Graphs

In this paper, we introduce the approach of graph densification as a means of preconditioning spectral clustering. After motivating the need of densification, we review the fundamentals of graph densifiers based on cut similarity and then analyze their associated optimization problems. In our experiments we analyze the implications of densification in the estimation of commute times.

Similarity Analysis from Limiting Quantum Walks

Similarity compression is a critical step to improve the efficiency of edge detection. In this paper, we compare two approaches for compressing/decompressing similarity matrices, being edge detection our application domain. In this regard, state-of-the-art contour detectors rely on spectral clustering where pixel or patch similarity is encoded in a symmetric weight matrix and the eigenvectors of the normalized Laplacian derived from this matrix are clustered in order to find contours (normalized cuts and its variants). Despite significant interest in learning the similarity measure for providing well localized boundaries, the underlying spectral analysis has played a subsidiary role, and has mostly been based on classical random walks and the heat kernel. However, recent findings based on continuous-time quantum walks suggest that under the complex wave equation there are long-range interactions not present in the classical case. In the case of the edge map this opens up a means of controlling texture in the edge map by a simple thresholding. In this paper, we use the long-time averages of quantum walks for edge detection, and show that texture is a consequence of short-rangedness of these interactions. This is due to the local-to-global property of limiting quantum walks. In addition, when analyzing the role of limiting quantum walks as intermediate/indirect similarity decompression, we find that quantum walks are able of recovering the original edge structure when a factorization compressor is used, whereas this is not the case when compression relies on the Szemeeredi Regularity Lemma, despite this latter method is by far more efficient.

Quantum vs Classical Ranking in Segment Grouping

In this paper we explore the use of ranking as a mean of guiding unsupervised image segmentation. Starting by the well known Pagerank algorithm we introduce an extension based on quantum walks. Pagerank rank can be used to prioritize the merging of segments embedded in uniform regions parts of the image with roughly similar appearance statistics. Quantum Pagerank, on the other hand, gives high priority to boundary segments. This latter effect is due to the higher order interactions captured by quantum fluctuations. However we found that qrank does not always outperform its classical version. We analyze the Pascal VOC database and give Intersection over Union IoU performances.