Lp Norm Multiple Kernel Fisher Discriminant Analysis for Object and Image Categorisation

Abstract

In this paper, we generalise multiple kernel Fisher discriminant analysis (MK-FDA) such that the kernel weights can be regularised with an ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> norm for any p ≥ 1, in contrast to existing MK-FDA that uses either l1 or l2 norm. We present formulations for both binary and multiclass cases and solve the associated optimisation problems efficiently with semi-infinite programming. We show on three object and image categorisation benchmarks that by learning the intrinsic sparsity of a given set of base kernels using a validation set, the proposed ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> MK-FDA outperforms its fixed-norm counterparts, and is capable of producing state-of-the-art performance. Moreover, we show that our ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> MK-FDA outperforms the ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> multiple kernel support vector machine (ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> MK-SVM) which has been recently proposed. Based on this observation and our experience with single kernel FDA and SVM, we argue that the almost century-old FDA is still a strong competitor of the popular SVM.