# What is "hard quantization" strategy?

I am working on classification and several times I encountered with this term. What is hard quantization strategy? What does it differ from soft approach?

Suppose you are given a data point $D$ and want to classify it into one of $N$ possible classes, $C_i$ for $i=1,2,\ldots, N$.

Hard classification chooses one of the $C_i$ to represent the classification of $D$.

Soft classification can be done in a number of ways, but the usual (for me) is to generate a weight $w_i$ for each of the $C_i$ which indicates the probability that the data point $D$ is part of the class $C_i$. This means that $$\sum_{i=1}^N w_i = 1$$.

I haven't heard the idea associated with quantization of one dimensional signals, but I've seen it used in relation to vector quantization which can be thought of as a simple classification technique.

This paper seems to have an example that matches what you say in your comment.

The idea is that, instead of each code word in the dictionary having the same weight, each code word has a weight dependent on some criterion or criteria: how far it is from other code words; how frequently it is likely to appear, etc. etc.

The picture below gives two examples (green and light blue) of data points that are better coded using a soft technique than a hard one.

• A similar idea applies for noisy signals in communications. An example: the receiver front-ends knows that $a$ is either +1 or -1, but observes $a+n$ where $n$ is random. It then outputs a "weight" indicating its degree of confidence in each of the two hypothesis.
– MBaz
Oct 31 '15 at 21:00
• @peterk, do you mean that in soft quantization we have weighting scheme for data points. more important, more weight? and in hard quantization, all data points have the same weights? Nov 1 '15 at 1:26
• In the case of digital receivers the hard quantization is given signal alphabet. For BPSK we have 2 members, for QPSK - 3, etc. By soft quantization we can understand soft metrics of demodulator. It could be (and probably it is) LLR metrics. They are calculated in the specific way according to maximum likelihood low and reflect the possibility of received point to be in some point of signal constellation. In one point of view it is weighting of received point. But such weighting is done not for each point of the alphabet but for neihbour one according to rule of calculation. There's no sense i
– Serj
Nov 2 '15 at 13:49
• CONT. We have assumption that noise dispersion not so much to place the point over neighboring position. If it is not, the signal is too corrupted to be recovered by any decoding scheme. It is for soft decoding. Of course in more common case we need to find all weights for any possible signal value. It is used in broodforce like schemes.
– Serj
Nov 2 '15 at 13:54
• @David :See my update.
– Peter K.
Nov 2 '15 at 14:43