Stable Magnitude Bottleneck

(1) Integration point: replace the usual linear latent projection in a small VAE, autoencoder, or contrastive encoder with a magnitude bottleneck. Given latent vector z in R^d, maintain m fixed random feature vectors r_j in R^d, where r_{j,i} is sampled independently from a centered distribution such as Rademacher, Gaussian, or a bounded symmetric distribution. Compute u = Rz, y = abs(u), and feed y to the decoder. Use a decoder that outputs z_hat or reconstructs x. The magnitude operation is the only nonlinearity in this module; do not apply independent learned sign flips after it, because the target ambiguity is one global sign.

(2) Pseudocode:

sample or initialize R[j,i] independently and centered
normalize each column so mean_j R[j,i]^2 = 1
for each minibatch x:
    z = encoder(x)                         # R^d
    u = z @ R.T                            # R^m
    y = abs(u)                             # magnitude bottleneck
    z_hat = decoder_latent(y)
    x_hat = decoder(z_hat)
    sigma = sign(sum_i z[i] * z_hat[i])    # stop gradient through sigma
    L_phase = mean(||z - sigma*z_hat||_2^2)
    L_recon = reconstruction_loss(x_hat, x)
    L = L_recon + lambda_phase*L_phase + beta*KL_or_prior_loss
    update network parameters

(3) The paper supplies the structural condition and the desired inequality, not a finite-sample value of C. Estimate each feature's empirical normalized L1 mass by a_i = mean_j abs(R[j,i]) / sqrt(mean_j R[j,i]^2), and reject or resample columns with a_i below a threshold such as 0.5. Estimate C_hat by generating pairs z and z + delta, decoding them, and taking the maximum observed ratio min_sigma ||z-sigma*z_hat||_2 / (||y-y_hat||_2+epsilon). Compare this against an unconstrained abs bottleneck using sparse, heavy-tailed, and near-zero feature distributions.

(4) First experiment: use MNIST or CIFAR-10 with a 16-dimensional latent autoencoder and m in {32, 64, 128}; compare a standard linear latent bottleneck, a magnitude bottleneck with Gaussian R, and the proposed L1-screened R. Add Gaussian noise to y at several signal-to-noise ratios and evaluate reconstruction error, latent recovery modulo global sign, and the empirical stability ratio C_hat. The expected signal is lower reconstruction degradation under magnitude noise, fewer catastrophic latent failures, and a monotone relationship between the minimum empirical a_i and robustness. A useful negative result is also falsifiable: if screened features do not improve C_hat or reconstruction at equal m, discard the method.