Sparse Representation Based Projections

original LPP SRLP LLE SRE Figure 1: SwissRoll 2D projections form 3D We adapt the Locality Preserving Projections (LPP)[4] and Locally Linear Embedding (LLE)[5] techniques to preserve the sparse representation property in the embedded space. The resulting techniques are Sparse Representation based Linear Projections (SRLP) and Sparse Representation based Embedding (SRE). We compare them to the original methods on several benchmarks, dealing with faces, traffic signs, digits, for both unsupervised and supervised cases. Most signals have a sparse representation as a linear combination of a reduced subset of signals from the same space naturally biased towards their own class. This is the starting point for Sparse Representation based Classification (SRC)[6]. The sparsity and compressed sensing idea brought new formulations such as Sparse PCA[7] or Sparse Regression Discriminant Analysis (SRDA)[1], which aim at having representations which are sparse over the basis directions in the embeddings. In an image-based recognition task we have a set of roughly aligned labeled training images {xi, li} from C classes. {xi ∈ RM} is the vectorial representation (here M grayscale pixel values), while li ∈ {1 . . .C} gives the class of the i-th image. We are searching a D-dimensional space such that the corresponding points {yi ∈ RD} preserve the sparse representation property as defined next. Let XN×M = [x1,x2, ...,xN ]T , YN×D = [y1,y2, ...,yN ]T , and N be the number of training samples. For each point xi we are searching for its sparse representation