# Euler-Hankel PSD Gram Activation

- ID: 2376
- Canonical URL: https://synthcore.org/idea/2376/euler-hankel-psd-gram-activation
- API JSON: https://synthcore.org/api/idea/2376.json
- API Markdown: https://synthcore.org/api/idea/2376.md
- Verification status: unverified
- Source: [arXiv:2608.15904](https://arxiv.org/abs/2608.15904)
- Category: architecture
- Solves: accuracy, stability, generalization
- ML areas: graph-nn, regularization, architecture
- Math tags: linear-algebra, convex-analysis, functional-analysis
- Ratings: usefulness 6/10; difficulty 5/10; novelty 7/10

## Idea description

Replace an unconstrained entrywise nonlinearity on a positive Gram or covariance matrix by a learned scalar function satisfying the paper's finite-order positivity-preserver conditions. The transformed matrix remains PSD for matrices of the target width n, allowing nonlinear Gram propagation without eigenvalue clipping or projection.

## Mathematical statement

For I=(0,\rho), let \mathbb{P}_n(I) be the cone of real n    imes n PSD matrices with entries in I. The paper defines the Euler operator \mathcal{E}=x\frac{d}{dx} and the Euler-Hankel matrix \mathcal{H}_n(f;x)=[\mathcal{E}^{i+j}f(x)]_{i,j=0}^{n-1}. For f\in C^{2n-2}(I), entrywise application f[A]=(f(a_{ij})) preserves PSD on \mathbb{P}_n(I) if and only if f^{(k)}(x)\geq0 for k=0,\ldots,n-1 and \mathcal{H}_n(f;x)\succeq0 for every x\in I. Here f^{(k)} is the ordinary kth derivative, \mathcal{E}^r is the r-fold Euler derivative, and \succeq0 means all eigenvalues are nonnegative. The implementation enforces these conditions on a dense grid covering the observed Gram-entry interval; this is an empirical approximation unless the constraints are established continuously.

## Key formulas

- $$f[A]=(f(a_{ij}))_{i,j=1}^{n},\qquad A\in\mathbb{P}_n(I),$$
- $$\mathcal{E}=x\frac{d}{dx},\qquad \mathcal{H}_{n}(f;x)=\bigl[\mathcal{E}^{i+j}f(x)\bigr]_{i,j=0}^{n-1},$$
- $$f[-]\text{ preserves positivity on }\mathbb{P}_{n}(I)\iff f^{(k)}(x)\geq0\ (0\leq k\leq n-1)\ \text{and}\ \mathcal{H}_{n}(f;x)\succeq0\ \forall x\in I,$$
- $$G=XX^{\mathsf T}+\epsilon I,\qquad G'_{ij}=f_\theta(G_{ij})$$

## Implementation notes

1. Integration point: add the layer after a feature matrix X\in\mathbb{R}^{b    imes d} has been converted to a positive Gram matrix G=XX^T+\epsilon I. If the batch dimension b is fixed, set n=b or use a smaller block size and apply the layer independently to blocks. Ensure entries are in I by normalizing G to a positive interval, for example G\leftarrow\operatorname{softplus}(G)+\delta, or use positive feature vectors so that G_{ij}>0. Apply f_    heta entrywise and feed the resulting PSD matrix to a graph message-passing or kernel readout.

2. Parameterize f_    heta by a small scalar MLP or polynomial basis on x\in[x_{\min},x_{\max}]. At every optimizer step, create a grid x_1,\ldots,x_m covering the observed entry range. Compute derivatives f_    heta^{(k)}(x_r) with autodiff for k=0,\ldots,n-1. Compute Euler derivatives recursively as E_0=f_    heta and E_{q+1}(x)=x\,dE_q(x)/dx, then form H_r[i,j]=E_{i+j}(x_r). Add the constraint loss L_c=\sum_{r,k}\operatorname{softplus}(-f^{(k)}(x_r))^2+\lambda\sum_r\operatorname{softplus}(    au-\lambda_{\min}(H_r))^2. Use a large penalty initially, then reduce it after the certificate margins become positive.

3. The paper supplies the certificate; the grid, autodiff derivatives, and minimum-eigenvalue computation are numerical engineering approximations. Monitor the minimum derivative and eigenvalue margins on a denser validation grid, and separately test random PSD matrices whose entries lie in I. Record the smallest eigenvalue of every transformed matrix, not just the constraint loss.

4. First experiment: compare a two-layer Gram network using ReLU, softplus, entrywise square, and the constrained f_    heta on MNIST or CIFAR-10 with block size n=16. Keep parameter count and optimizer fixed. Measure classification accuracy, frequency and magnitude of negative eigenvalues, and training stability. Success is comparable or better accuracy with zero PSD violations up to numerical tolerance and fewer instability events than unconstrained activations.

## Disclaimer

AI-generated research hypothesis, automatically tested. Not peer-reviewed.