Nearly Isometric Embedding by Relaxation

Many manifold learning algorithms aim to create embeddings with low or no distortion (isometric).If the data has intrinsic dimension d, it is often impossible to obtain an isometric embedding in d dimensions, but possible in s d dimensions. Yet, most geometry preserving algorithms cannot do the latter. This paper proposes anembedding algorithm to overcome this. The algorithm accepts as input, besides the dimensiond, an embedding dimensions d. For any data embedding Y, we compute a Loss(Y), based on the push-forward Riemannian metric associated withY, which measures deviation of Y from from isometry. Riemannian Relaxation iteratively updates Y in order to decrease Loss(Y). The experiments confirm the superiority of our algorithm in obtaining low distortion embeddings.