Variable Selection for K - Means Quantization

Recent results in quantization theory provide theoretical bounds on the distortion of squared-norm based quantizers (see, e.g., [3] or [10]). These bounds are valid whenever the source distribution has a bounded support, regardless of the dimension of the underlying Hilbertian space. However, it remains of interest to select relevant variable for quantization. This task is usually performed using coordinate energy-ratio thresholding (see, e.g., [1] or [17]), or maximizing a constrained empirical Between Cluster Sum of Squares criterion (see, e.g., [4] or [22]). This paper offers a Lasso type procedure to select the relevant variables for k-means clustering, as exposed in [18]. Moreover, some non-asymptotic convergence results on the distortion are derived for this procedure, along with consistency results toward sparse codebooks.