Manifold models for signals and images
Computer Vision and Image Understanding
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PCS'09 Proceedings of the 27th conference on Picture Coding Symposium
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UAI '09 Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence
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IEEE Transactions on Signal Processing
Joint manifolds for data fusion
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IEEE Transactions on Signal Processing
Two-dimensional random projection
Signal Processing
Compressive evaluation in human motion tracking
ACCV'10 Proceedings of the 10th Asian conference on Computer vision - Volume Part IV
Trading spaces: on the lore and limitations of latent semantic analysis
ICTIR'11 Proceedings of the Third international conference on Advances in information retrieval theory
Dimensionality reduction: beyond the Johnson-Lindenstrauss bound
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
The Journal of Machine Learning Research
Target reconstruction using manifold-based compressive sensing
IScIDE'11 Proceedings of the Second Sino-foreign-interchange conference on Intelligent Science and Intelligent Data Engineering
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Proceedings of the 27th Conference on Image and Vision Computing New Zealand
Distance preserving embeddings for general n-dimensional manifolds
The Journal of Machine Learning Research
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We propose a new approach for nonadaptive dimensionality reduction of manifold-modeled data, demonstrating that a small number of random linear projections can preserve key information about a manifold-modeled signal. We center our analysis on the effect of a random linear projection operator Φ:ℝ N →ℝM , MN, on a smooth well-conditioned K-dimensional submanifold ℳ⊂ℝN . As our main theoretical contribution, we establish a sufficient number M of random projections to guarantee that, with high probability, all pairwise Euclidean and geodesic distances between points on ℳ are well preserved under the mapping Φ. Our results bear strong resemblance to the emerging theory of Compressed Sensing (CS), in which sparse signals can be recovered from small numbers of random linear measurements. As in CS, the random measurements we propose can be used to recover the original data in ℝN . Moreover, like the fundamental bound in CS, our requisite M is linear in the “information level” K and logarithmic in the ambient dimension N; we also identify a logarithmic dependence on the volume and conditioning of the manifold. In addition to recovering faithful approximations to manifold-modeled signals, however, the random projections we propose can also be used to discern key properties about the manifold. We discuss connections and contrasts with existing techniques in manifold learning, a setting where dimensionality reducing mappings are typically nonlinear and constructed adaptively from a set of sampled training data.