Learning sparse filter bank transforms with convolutional ICA

Independent Component Analysis (ICA) is a generalization of Principal Component Analysis that optimizes a linear transformation to whiten and sparsify a family of source signals. The computational costs of ICA grow rapidly with dimensionality, and application to high-dimensional data is generally achieved by restricting to small windows, violating the translation-invariant nature of many real-world signals, and producing blocking artifacts in applications. Here, we reformulate the ICA problem for transformations computed through convolution with a bank of filters, and develop a generalization of the fastICA algorithm for optimizing the filters over a set of example signals. This results in a substantial reduction of computational complexity and memory requirements. When applied to a database of photographic images, the method yields bandpass oriented filters, whose responses are sparser than those of orthogonal wavelets or block DCT, and slightly more heavy-tailed than those of block ICA, despite fewer model parameters.