Potential-weighted fractional diffusion layer
Implementation & benchmark of arXiv:2608.17406 — Potential-free $L^1$-estimates for positivity-preserving Riesz transform related to Schrödinger operator in dimension one
Source paper: Potential-free $L^1$-estimates for positivity-preserving Riesz transform related to Schrödinger operator in dimension one arXiv:2608.17406 ⓘ · analyzed Sep 1, 2026
AI-generated research hypothesis, automatically tested. Not peer-reviewed.
Idea description
Insert a positivity-preserving fractional Schrödinger resolvent into a 1D neural sequence block. Given a nonnegative learned potential V, the layer transforms an input signal f using V^a(-Delta+V)^(-a)f, allowing the network to learn where to smooth or suppress features while retaining an L1 bound independent of the potential magnitude. Use a in (0,1] as a fixed hyperparameter or a clipped learned scalar.
Formulas
Mathematical statement
The paper studies the Schrödinger operator L=-Delta+V, where in one dimension -Delta=-d^2/dx^2 and V is a locally integrable nonnegative potential. Its central result states that for every 0<a<=1, there is a constant C_a independent of V such that ||V^a L^(-a) f||_{L^1(R)} <= C_a ||f||_{L^1(R)}. Here f is the input signal, L^(-a) is the negative fractional power of the positive operator L, and V^a is pointwise multiplication by V(x)^a. The transfer uses a finite-grid approximation H=-D_2+diag(v+epsilon), where D_2 is the discrete second-derivative operator, v_i>=0 is a network-predicted potential, and epsilon>0 avoids singularities at zero potential. The fractional inverse can be computed through the Balakrishnan heat-semigroup identity H^(-a)=Gamma(a)^(-1) integral from 0 to infinity of t^(a-1) exp(-tH) dt. The theorem motivates monitoring the discrete weighted L1 gain ||diag(v+epsilon)^a H^(-a)f||_1 / ||f||_1, although the continuum theorem does not automatically provide the same constant for every discretization or boundary condition.
Implementation notes
(1) Integration point: apply this layer to a length-n feature channel or to each channel of a 1D convolution or state-space block, before residual addition. Let x in R^(n times c) be the feature sequence. A small pointwise MLP predicts v=softplus(g(x averaged over channels))+v_min in R^n. Construct a tridiagonal Neumann or Dirichlet finite-difference matrix D_2 with grid spacing h, and set H=-D_2+diag(v+epsilon). (2) Pseudocode: v = softplus(mlp(mean_channels(x))) + v_min; H = -D2 + diag(v+eps); z = fractional_inverse(H,a,x); y = (v+eps)^a elementwise multiplied by z; return x + alpha times norm(y). The fractional_inverse operation can use eigendecomposition for n<=256, a Lanczos or Krylov approximation to H^(-a)x, or quadrature of the heat identity using matrix exponentials. Learn a=sigmoid(a_raw) constrained to [0.05,1], or begin with a=0.5. (3) Compute exactly the tridiagonal Schrödinger matrix, the fractional inverse approximation, and the pointwise multiplication. Estimate approximation error by comparing Lanczos output with eigendecomposition on short sequences. Estimate empirical L1 gain using random nonnegative x and the ratio sum(abs(y))/sum(abs(x)). Do not assume the continuum constant C_a automatically holds on the grid. (4) First experiment: train a small 1D U-Net or S4-style sequence model on noisy-signal denoising and ECG classification. Compare an identity residual block, ordinary Gaussian or learned diffusion, and this layer at equal parameter count and FLOPs. Sweep potentials from smooth to sharply varying and measure gradient norms, activation L1 gain, corruption robustness, and validation accuracy. Success means no exploding L1 gain as potential amplitude increases, fewer NaNs or gradient spikes, and improved denoising or classification at matched compute.
Verification
This idea has not been verified yet.
Verification happens in two stages: Stage 1 — a mechanism check on a toy system confirms the claimed mathematical phenomenon reproduces; Stage 2 — a benchmark implements the idea on a real (small) neural network task and compares it against a tuned baseline over 8 paired seeds with a permutation test.
Artifacts
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