Non-negativity constraints on the pre-image for pattern recognition with kernel machines

• The pre-image problem for pattern recognition.
• We study a constrained pre-image problem with non-negativity constraints.
• New theoretical results on the pre-image problem, including conditions for the convexity of the preimage problem.
• A fortuitous side-effect of our method is the sparsity in the representation, a property investigated in this paper.

Rules of physics in many real-life problems force some constraints to be satisfied. This paper deals with nonlinear pattern recognition under non-negativity constraints. While kernel principal component analysis can be applied for feature extraction or data denoising, in a feature space associated to the considered kernel function, a pre-image technique is required to go back to the input space, e.g., representing a feature in the space of input signals. The main purpose of this paper is to study a constrained pre-image problem with non-negativity constraints. We provide new theoretical results on the pre-image problem, including the weighted combination form of the pre-image, and demonstrate sufficient conditions for the convexity of the problem. The constrained problem is considered with the non-negativity, either on the pre-image itself or on the weights. We propose a simple iterative scheme to incorporate both constraints. A fortuitous side-effect of our method is the sparsity in the representation, a property investigated in this paper. Experimental results are conducted on artificial and real datasets, where many properties are investigated including the sparsity property, and compared to other methods from the literature. The relevance of the proposed method is demonstrated with experimentations on artificial data and on two types of real datasets in signal and image processing.

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