Unverified 2026

Injective Boundary-Aware Disk Pooling

Implementation & benchmark of arXiv:2608.02546 — Variable-Radius Disk Transforms and an Area-Integral Problem of Zalcman

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Architecture Accuracy Stability Sample efficiency Cnn Representation Regularization Functional analysis Geometry Measure theory

Source paper: Variable-Radius Disk Transforms and an Area-Integral Problem of Zalcman arXiv:2608.02546 ⓘ · analyzed Aug 31, 2026

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 Not peer-reviewed.

Idea description

Replace fixed-radius image blur or pooling with disk averages whose radius is proportional to the distance from each pixel to the image boundary. Compute the transform at every spatial location and train a lightweight decoder to reconstruct the pre-transform feature map, using reconstruction error as an anti-collapse regularizer. This creates a scale-adaptive smoothing layer with an injectivity motivation in the continuum while providing larger context in the image interior.

Formulas

$$(\mathcal T_\alpha f)(z)=\int_{B(z,\alpha(1-|z|))}f(\zeta)\,dA(\zeta),\qquad B(z,r)=\{\zeta\in\mathbb C:|\zeta-z|<r\}.$$

$$0<\alpha<1,\ f\in C(\mathbb D)\cap L^\infty(\mathbb D):\quad \mathcal T_\alpha f=\mathcal T_\alpha g\ \Longrightarrow\ f=g;\qquad \alpha=1,\ f\in L^1(\mathbb D):\quad \mathcal T_1 f=\mathcal T_1 g\ \Longrightarrow\ f=g.$$

$$P_\alpha f(x_i)=\frac{1}{\pi r_i^2}\int_{B(x_i,r_i)}f(\zeta)\,dA(\zeta),\qquad r_i=\alpha(1-|x_i|),$$

$$\mathcal L=\mathcal L_{task}(y,\hat y)+\lambda_{rec}\frac{1}{CHW}\|D_\phi(P_\alpha(F))-F\|_2^2.$$

Mathematical statement

For the unit disk $\mathbb D=\{z\in\mathbb C:|z|<1\}$, define $B(z,r)=\{\zeta\in\mathbb C:|\zeta-z|<r\}$ and $(\mathcal T_\alpha f)(z)=\int_{B(z,\alpha(1-|z|))}f(\zeta)\,dA(\zeta)$, where $0<\alpha\leq1$, $dA$ is planar Lebesgue measure, and $f$ is a scalar or channelwise feature field. The paper proves that $\mathcal T_\alpha$ is injective on $C(\mathbb D)\cap L^\infty(\mathbb D)$ for $0<\alpha<1$, and injective on $L^1(\mathbb D)$ for $\alpha=1$: equality of all disk integrals implies equality of the underlying functions in the stated class. For neural features, map pixel coordinates $x_i$ into $\mathbb D$, set $r_i=\alpha(1-|x_i|)$, and use the normalized transform $P_\alpha f(x_i)=(\pi r_i^2)^{-1}\int_{B(x_i,r_i)}f(\zeta)dA(\zeta)$; normalization changes scale but not injectivity. A finite grid is not guaranteed exactly injective because of sampling, so the reconstruction term empirically encourages a well-conditioned discrete approximation.

Implementation notes

Integrate this at the first or second CNN stage, immediately after a feature tensor $F\in\mathbb R^{B\times C\times H\times W}$ and before stride-2 pooling. Normalize pixel centers as $x_{ij}=(2j/(W-1)-1,2i/(H-1)-1)$, multiply coordinates by $0.99$ so all centers lie strictly inside $\mathbb D$, and compute $r_{ij}=\alpha(1-\|x_{ij}\|_2)$. Approximate each disk integral with a fixed antialiased stencil of 16--32 offsets $u_k$ satisfying $\|u_k\|_2\leq1$: r = alpha * (1 - norm(coords)); P = 0; Z = 0; for (u,w) in stencil: P += w * grid_sample(F, coords + r*u); Z += w; P /= Z. Use bilinear grid_sample so the layer is differentiable and cache coordinate grids for each resolution. Feed $P$ into the original network and add a small three-layer 1x1 decoder $D_\phi$ with loss $\lambda_{rec}\operatorname{mean}((D_\phi(P)-F)^2)$. The paper supplies the continuum transform and injectivity theorem; estimate discrete conditioning by explicitly constructing the single-channel $HW\times HW$ matrix on a small grid and measuring its smallest singular value or condition number. Tune $\alpha\in\{0.25,0.5,0.75,1.0\}$, minimum radius, and $\lambda_{rec}\in\{0,10^{-4},10^{-3},10^{-2}\}$. First test on CIFAR-10 with ResNet-18, comparing fixed 3x3 average pooling, Gaussian blur, and this transform at equal FLOPs. Record validation accuracy, corruption robustness, reconstruction MSE, gradient norms, and boundary-versus-interior errors. Success means improved accuracy or robustness at matched compute, lower reconstruction error than fixed pooling, and no severe condition-number explosion near the boundary.

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.

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