| Robust PCA by Convex Optimization |
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Classical PCA is a beautiful algorithm, with applications
from vision and signal processing to bioinformatics and search and
ranking. Unfortunately, while PCA is optimal for recovering a low-rank
matrix under small Gaussian noise, it breaks down under
highly-corrupted or outlying observations. Fortunately, for most cases, the problem of recovering such a low-rank matrix A from highly-corrupted observations D = A + E can be efficiently and exactly solved by convex optimization: A natural semidefinite
programming relaxation perfectly
recovers almost all d x d
matrices A of rank O(d/log d) from almost all errors E affecting cd2 of the observations.
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| Face Recognition by Sparse Representation: Theory and Practice |
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Automatic face recognition is a classical problem in computer
vision. Despite much progress over the last 30 years, making
recognition systems work in the real world is still a surprisingly
difficult challenge. This project shows how face recognition under varying illumination, occlusion and alignment error can be effectively cast as a sparse representation problem, which can then be solved using tools from convex programming. In turn, peculiarities of face images reveal surprising phenomena in the theory of sparse representation, including the correction of large fractions of errors and the success of l1 minimization even in highly coherent dictionaries that violate classical sufficient conditions such as the restricted isometry property. Read more: Robust Face Recognition via Sparse Representation Towards a Practical Automatic Face Recognition System: Robust Illumination and Alignment by Sparse Representation Dense Error Correction via L1 Minimization Project webpage - Face Recognition by Sparse Representation |
| Clustering and Classification by Lossy Data Compression |
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This project investigates the use of lossy data coding and compression as
a tool for clustering and classifying multivariate data.
Compression provides a natural answer to the model selection problem,
and also has nice regularization effects in classification. Algorithms built around these principles perform very effectively on real-world tasks involving mixture distributions with components of varying dimension. One important example of this is natural image segmentation, where the allowable distortion in computing the coding length provides a natural measure of scale. Read more: Segmentation of Multivariate Mixed Data via Lossy Coding and Compression Classification via Minimum Incremental Coding Length (MICL) Unsupervised Segmentation of Natural Images via Lossy Data Compression Project webpage - Clustering and Classification by Lossy Compression |