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Classical quantization technique from signal processing

Vector quantization (VQ) is a classical quantization technique from signal processing that allows the modeling of probability density functions by the distribution of prototype vectors. Developed in the early 1980s by Robert M. Gray, it was originally used for data compression. It works by dividing a large set of points (vectors) into groups having approximately the same number of points closest to them. Each group is represented by its centroid point, as in k means and some other clustering algorithms. In simpler terms, vector quantization chooses a set of points to represent a larger set of points. The density matching property of vector quantization is powerful, especially for identifying the density of large and high dimensional data. Since data points are represented by the index of their closest centroid, commonly occurring data have low error, and rare data high error. This is why VQ is suitable for lossy data compression. It can also be used for lossy data correction and density estimation. Vector quantization is based on the competitive learning paradigm, so it is closely related to the self organizing map model and to sparse coding models used in deep learning algorithms such as autoencoder.

Vector quantization is used for lossy data compression, lossy data correction, pattern recognition, density estimation and clustering. Lossy data correction, or prediction, is used to recover data missing from some dimensions. It is done by finding the nearest group with the data dimensions available, then predicting the result based on the values for the missing dimensions, assuming that they will have the same value as the group's centroid. For density estimation, the area/volume that is closer to a particular centroid than to any other is inversely proportional to the density (due to the density matching property of the algorithm).

Vector quantization, also called "block quantization" or "pattern matching quantization" is often used in lossy data compression. It works by encoding values from a multidimensional vector space into a finite set of values from a discrete subspace of lower dimension. A lower space vector requires less storage space, so the data is compressed. Due to the density matching property of vector quantization, the compressed data has errors that are inversely proportional to density. The transformation is usually done by projection or by using a codebook. In some cases, a codebook can be also used to entropy code the discrete value in the same step, by generating a prefix coded variable length encoded value as its output. The set of discrete amplitude levels is quantized jointly rather than each sample being quantized separately. Consider a k dimensional vector of amplitude levels. It is compressed by choosing the nearest matching vector from a set of n dimensional vectors , with n < k.

All possible combinations of the n dimensional vector form the vector space to which all the quantized vectors belong. Only the index of the codeword in the codebook is sent instead of the quantized values. This conserves space and achieves more compression. Twin vector quantization (VQF) is part of the MPEG 4 standard dealing with time domain weighted interleaved vector quantization.

As VQ is seeking for centroids as density points of nearby lying samples, it can be also directly used as a prototype based clustering method: each centroid is then associated with one prototype. By aiming to minimize the expected squared quantization error and introducing a decreasing learning gain fulfilling the Robbins Monro conditions, multiple iterations over the whole data set with a concrete but fixed number of prototypes converges to the solution of k means clustering algorithm in an incremental manner.

VQ has been used to quantize a feature representation layer in the discriminator of Generative adversarial networks. The feature quantization (FQ) technique performs implicit feature matching. It improves the GAN training, and yields an improved performance on a variety of popular GAN models: BigGAN for image generation, StyleGAN for face synthesis, and U GAT IT for unsupervised image to image translation.