Logarithmic Random-Hyperplane Classifier

Integrate this at the final classifier: let the backbone produce an embedding $h\in\mathbb{R}^d$, then apply a binary projection matrix $R\in\mathbb{R}^{N\times d}$ instead of a $K\times d$ softmax weight matrix. Set $N=\lceil2\log_2K+\log_2(1/\delta)\rceil$ for an initial trial, sample each row of $R$ from an isotropic Gaussian and normalize it, and keep $R$ fixed during the first experiment. Generate one class codeword per class by evaluating $\operatorname{sign}(R\mu_k)$ on class prototype embeddings $\mu_k$, or use a fixed balanced random codebook with rejection sampling until all codewords are distinct. During training, compute logits $s=Rh$, use the target signs $c_y$, and optimize $\sum_{n=1}^N\operatorname{softplus}(-c_{y,n}s_n)$; optionally add a small norm penalty on $h$. At inference, use either hard Hamming decoding or soft distance $\arg\min_k\sum_n\operatorname{softplus}(-c_{k,n}s_n)$. The formula for $N$ is taken from the paper; code collisions, bit balance, and empirical class margins are measured directly. First test on CIFAR-100 and CIFAR-100-LT with a small ResNet, comparing against a standard softmax at matched trainable classifier parameters and matched FLOPs. Success means comparable top-1 accuracy with fewer final-layer parameters, lower communication cost, or faster inference; failure is a substantial accuracy drop after tuning the embedding dimension.