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Compressive learning forms the exciting intersection between compressed sensing and statistical learning where one exploits sparsity of the learning model to reduce the memory and/or computational complexity of the algorithms used to solve the learning task. In this paper, we look at the independent component analysis (ICA) model through the compressive learning lens. In particular, we show that solutions to the cumulant-based ICA model have a particular structure that induces a low-dimensional model set that resides in the cumulant tensor space. By showing that a restricted isometry property holds for random cumulants e.g. Gaussian ensembles, we prove the existence of a compressive ICA scheme. Thereafter, we propose two algorithms of the form of an iterative projection gradient and an alternating steepest descent algorithm for compressive ICA, where the order of compression asserted from the restricted isometry property is realized through empirical results. We provide analysis of the CICA algorithms including the effects of finite samples. The effects of compression are characterized by a trade-off between the sketch size and the statistical efficiency of the ICA estimates. By considering synthetic and real datasets, we show the substantial memory gains achieved over well-known ICA algorithms by using one of the proposed CICA algorithms. [ABSTRACT FROM AUTHOR]

Recently, many works have focused on the characterization of non-linear dimensionality reduction methods obtained by quantizing linear embeddings, e.g., to reach fast processing time, efficient data compression procedures, novel geometry-preserving embeddings or to estimate the information/bits stored in this reduced data representation. In this work, we prove that many linear maps known to respect the restricted isometry property (RIP) can induce a quantized random embedding with controllable multiplicative and additive distortions with respect to the pairwise distances of the data points beings considered. In other words, linear matrices having fast matrix-vector multiplication algorithms (e.g., based on partial Fourier ensembles or on the adjacency matrix of unbalanced expanders) can be readily used in the definition of fast quantized embeddings with small distortions. This implication is made possible by applying right after the linear map an additive and random "dither" that stabilizes the impact of the uniform scalar quantization operator applied afterwards. For different categories of RIP matrices, i.e., for different linear embeddings of a metric space (K⊂Rn,ℓq) in (Rm,ℓp) with p,q≥1, we derive upper bounds on the additive distortion induced by quantization, showing that it decays either when the embedding dimension m increases or when the distance of a pair of embedded vectors in K decreases. Finally, we develop a novel "bi-dithered" quantization scheme, which allows for a reduced distortion that decreases when the embedding dimension grows and independently of the considered pair of vectors. [ABSTRACT FROM AUTHOR]