How to represent +1 and -1 in one bit?

[airjimmy89]

As mentioned in the BinaryNet paper, we can use the XNOR-Bitcounting operation. It seems to be reasonable, but when we really try to realize this in one bit, how should we represent -1? Should we convert it into 0? And that seems to meet some problems when we try to xnor it with the input 8-bit vector, since mostly a negative number will be in 2-complement form.. I might be wrong, just a little bit confused since I was wondering how to achieve this on hardware...

[MatthieuCourbariaux]

It seems to be reasonable, but when we really try to realize this in one bit, how should we represent -1? Should we convert it into 0?

In practice, we use 0s to represent the -1s, XOR-popcount-adds to compute the weighted sums, and finally we adjust these weighted sums so that we get the same result as a dot-product (provided the operands are constrained to (-1,1)).

And that seems to meet some problems when we try to xnor it with the input 8-bit vector, since mostly a negative number will be in 2-complement form..

We explain in the paper (Section 1.6) how to handle 8-bit inputs as shifted sums of binary inputs.

[airjimmy89]

Thanks for the reply! :) Is it because the input pixel value is an 8bit uint number that we can use the XOR-popcount-adds method? It does get a right answer with unsigned input vector, but how about a vector of signed numbers (maybe we do not need to consider such situation?)?

[airjimmy89]

And what if one of the weight vector is all 1s? Overflowed? In case of that, we should take a (8+log2(vectorlength))bit number to represent the output?