My research interests span a relatively wide area, ranging from Signal Processing to Matrix Analysis, and from Algorithms to Numerical Optimization. In my work, the main focus is on deriving adaptive algorithms for efficient representations of natural signals (e.g., images, speech sound), and on investigating their theoretical properties (such as computational complexity and accuracy).

• Subgraph-Preconditioned Conjugate Gradient. A central problem in robot navigation - simultaneous localization and mapping (SLAM) - can be formulated as inference in a factor graph. We solve the associated least-squares optimization via a preconditioned conjugate gradient of linear system, by choosing a subgraph preconditioner guaranteed to better condition the problem. Our method nontrivially extends and adapts recent algorithmic tools in theoretical computer science, specifically low-stretch trees and ultrasparsifiers. Moreover, to assess the quality of the preconditioners, we generalize existing linear algebraic tools developed for subgraph preconditioning (specifically support theory) to the case of general, non-scalar factor graphs.
  
  
• Convergence Behavior of some specialized Cellular Automata. By recasting the newly proposed Active Mask image segmentation algorithm in a cellular automata framework, we may facilitate the analysis and theoretical guarantees on the convergence of this voting-based iterative algorithm to fixed-point (zero-change) states. For examples and proofs, check out our paper. An earlier version appeared in IEEE ICASSP 2010.
  
  
• Spike-based representations. For temporal signals, look here for details. Currently, I am studying the visual flavor of this approach, that is, I am working on the derivation of an adaptive, shiftable-kernel, highly sparse representation for images. The main computational advantage of this approach lies in a very efficient implementation of the Matching Pursuit algorithm used for spike extraction in the inference stage, and on superfast optimization of the dictionary in the learning stage.
  
  
• Robustness and redundancy in linear representations. "Robust Coding", or "how to cope with noise in the (linear) representation by using more coding units". For more on this (and for nice pictures), check out our paper in IEEE Trans. on Image Processing. A very intuitive and natural characterization of the optimal code in the two-dimensional case appeared in a preliminary version in NIPS 2005.
  
  
• Speech Enhancement by Bandwidth Extension. Reconstructing speech with noisy or missing spectral content appears in digital telephony or blind source separation from mono- or multi-channel data. By modeling this problem as statistical inference with missing data, we can restore altered speech frames based on similarity with clean ones. Generality and speed are due to only using informative training examples in the partial matching stage, while reconstruction quality both perceptual and quantitative (by the Itakura-Saito criterion), is at least as good as that of specialized methods. For more details, look here and here.
  
  
• Wavelets, frames, filter-banks. See how to increase coding efficiency of multiscale/multiresolution methods for a given ensemble of images, by employing statistical adaptivity of the representation. Here is our paper on Multiresolution ICA (MrICA) in IEEE ICASSP 2009. Our combined multiresolution adaptive approach performs better than JPEG2000 on several classes of images, by learning a better set of features for each of these.
  
  
• Algebraic Signal Processing Theory. For more on what that is, please see this source. If you want to see how to design alternative polynomial transforms that asymptotically approach the DTFT (their spectrum converges to the unit circle) but avoid the periodic signal extension/boundary condition, see our paper. If you still want to see more, check out this video of me at the NIPS Workshop "Algebraic and combinatorial methods in machine learning".