Scalar Quantization as Sparse Least Square Optimization

Quantization can be used to form new vectors/matrices with shared values close to the original. In recent years, the popularity of scalar quantization for value-sharing applications has been soaring as it has been found huge utilities in reducing the complexity of neural networks. Existing clustering-based quantization techniques, while being well-developed, have multiple drawbacks including the dependency of the random seed, empty or out-of-the-range clusters, and high time complexity for a large number of clusters. To overcome these problems, in this paper, the problem of scalar quantization is examined from a new perspective, namely sparse least square optimization. Specifically, inspired by the property of sparse least square regression, several quantization algorithms based on $l_1$ least square are proposed. In addition, similar schemes with $l_1 + l_2$ and $l_0$ regularization are proposed. Furthermore, to compute quantization results with a given amount of values/clusters, this paper designed an iterative method and a clustering-based method, and both of them are built on sparse least square. The paper shows that the latter method is mathematically equivalent to an improved version of k-means clustering-based quantization algorithm, although the two algorithms originated from different intuitions. The algorithms proposed were tested with three types of data and their computational performances, including information loss, time consumption, and the distribution of the values of the sparse vectors, were compared and analyzed. The paper offers a new perspective to probe the area of quantization, and the algorithms proposed can outperform existing methods especially under some bit-width reduction scenarios, when the required post-quantization resolution (number of values) is not significantly lower than the original number.