Publications

Audio inpainting with similarity graphs

I’m very proud to announce the release of a new kind of audio inpainting algorithm. It is able to reconstruct long missing parts of a song by searching through the rest of the content for a suitable replacement.

You can try the algorithm online or download the code here and run it on your machine. A technical report associated with this algorithm is available on arXiv.

Enjoy!

Abstract
In this contribution, we present a method to compensate for long duration data gaps in audio signals, in particular music. To achieve this task, a similarity graph is constructed, based on a short-time Fourier analysis of reliable signal segments, e.g. the uncorrupted remainder of the music piece, and the temporal regions adjacent to the unreliable section of the signal. A suitable candidate segment is then selected through an optimization scheme and smoothly inserted into the gap.

New paper: Global and Local Uncertainty Principles for Signals on Graphs

I’m particularly proud to present this work since we have been working on it for almost 4 years. It started with my master thesis in 2012 before any publications on graph uncertainty were out. Even though there were new coming articles dealing with uncertainty every few months, we kept working our initial idea.

In this paper, we generalize some classical uncertainty principles for signals residing on Euclidean domains to uncertainty principles for signals residing on weighted graphs. To do so, we use generalizations of time-frequency transforms and ambiguity functions. Contrary to the classical setting, the uncertainty in the graph setting depends on the localization of the signal, leading to the new concept of “local uncertainty.”

ArXiv link: http://arxiv.org/abs/1603.03030

Abstract

Uncertainty principles such as Heisenberg’s provide limits on the time-frequency concentration of a signal, and constitute an important theoretical tool for designing and evaluating linear signal transforms. Generalizations of such principles to the graph setting can inform dictionary design for graph signals, lead to algorithms for reconstructing missing information from graph signals via sparse representations, and yield new graph analysis tools. While previous work has focused on generalizing notions of spreads of a graph signal in the vertex and graph spectral domains, our approach is to generalize the methods of Lieb in order to develop uncertainty principles that provide limits on the concentration of the analysis coefficients of any graph signal under a dictionary transform whose atoms are jointly localized in the vertex and graph spectral domains. One challenge we highlight is that due to the inhomogeneity of the underlying graph data domain, the local structure in a single small region of the graph can drastically affect the uncertainty bounds for signals concentrated in different regions of the graph, limiting the information provided by global uncertainty principles. Accordingly, we suggest a new way to incorporate a notion of locality, and develop local uncertainty principles that bound the concentration of the analysis coefficients of each atom of a localized graph spectral filter frame in terms of quantities that depend on the local structure of the graph around the center vertex of the given atom. Finally, we demonstrate how our proposed local uncertainty measures can improve the random sampling of graph signals.

New paper: compressive PCA on graphs

http://arxiv.org/abs/1602.02070

Abstract

Randomized algorithms reduce the complexity of low-rank recovery methods only w.r.t dimension p of a big dataset YRp×n. However, the case of large n is cumbersome to tackle without sacrificing the recovery. The recently introduced Fast Robust PCA on Graphs (FRPCAG) approximates a recovery method for matrices which are low-rank on graphs constructed between their rows and columns. In this paper we provide a novel framework, Compressive PCA on Graphs (CPCA) for an approximate recovery of such data matrices from sampled measurements. We introduce a RIP condition for low-rank matrices on graphs which enables efficient sampling of the rows and columns to perform FRPCAG on the sampled matrix. Several efficient, parallel and parameter-free decoders are presented along with their theoretical analysis for the low-rank recovery and clustering applications of PCA. On a single core machine, CPCA gains a speed up of p/k over FRPCAG, where k << p is the subspace dimension. Numerically, CPCA can efficiently cluster 70,000 MNIST digits in less than a minute and recover a low-rank matrix of size 10304 X 1000 in 15 secs, which is 6 and 100 times faster than FRPCAG and exact recovery.

New paper: Stationary signal processing on graphs

I’m proud to present a new paper. Using the ideas presented inside, we should be able to improve many graph-based models.

Abstract

Graphs are a central tool in machine learning and information processing as they allow to conveniently capture the structure of complex datasets. In this context, it is of high importance to develop flexible models of signals defined over graphs or networks. In this paper, we generalize the traditional concept of wide sense stationarity to signals defined over the vertices of arbitrary weighted undirected graphs. We show that stationarity is intimately linked to statistical invariance under a localization operator reminiscent of translation. We prove that stationary graph signals are characterized by a well-defined Power Spectral Density that can be efficiently estimated even for large graphs. We leverage this new concept to derive Wiener-type estimation procedures of noisy and partially observed signals and illustrate the performance of this new model for denoising and regression.

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