Laplacian Eigenmaps from Sparse, Noisy Similarity Measurements
This addresses computational efficiency and robustness in manifold learning for applications like speech processing and neuroscience, but is incremental as it builds on existing Laplacian eigenmaps methods.
The paper tackles the problem of how noise and occlusion in similarity measurements affect Laplacian eigenmaps embeddings, showing that a good approximation can be recovered with high probability under modest assumptions, and explores this with real-world and synthetic datasets.
Manifold learning and dimensionality reduction techniques are ubiquitous in science and engineering, but can be computationally expensive procedures when applied to large data sets or when similarities are expensive to compute. To date, little work has been done to investigate the tradeoff between computational resources and the quality of learned representations. We present both theoretical and experimental explorations of this question. In particular, we consider Laplacian eigenmaps embeddings based on a kernel matrix, and explore how the embeddings behave when this kernel matrix is corrupted by occlusion and noise. Our main theoretical result shows that under modest noise and occlusion assumptions, we can (with high probability) recover a good approximation to the Laplacian eigenmaps embedding based on the uncorrupted kernel matrix. Our results also show how regularization can aid this approximation. Experimentally, we explore the effects of noise and occlusion on Laplacian eigenmaps embeddings of two real-world data sets, one from speech processing and one from neuroscience, as well as a synthetic data set.