arXiv subjects

Hy P. G. Lam

Publications and source records attributed to Hy P. G. Lam.

5 recordsLinked to original sources

arXiv2026

Sharp Reconstruction Bounds for Autoencoders Using the Same Forward Map

We study reconstruction in autoencoders that apply the same forward map before and after setting the observed coordinates to zero. For equal odd input and hidden dimensions $d\geq 3$, among orientation-preserving diffeomorphisms whose Jacobian singular values lie in $[m,M]$, we show that the least uniform reconstruction-derivative error is $\max\{1-M(M-m)/2,0\}$, with affine maps attaining this sharp bound at every prescribed depth. A translated radial rotation can nevertheless reconstruct any prescribed ball exactly with singular values arbitrarily close to one, motivating additional conditions for a finite-data bound. We test this prediction on a 798,452-point terrestrial LiDAR forest scan. At input scale $0.05$, the mean theoretical bound is $0.155$, about $84\%$ of the mean normalized training error $0.185$ across four spatial regions, two depths, and three seeds. At this scale, adding one hidden coordinate reduces the mean reconstruction error below $6\times10^{-6}$.

cs.LG↗

arXiv2026

Zero-energy scattering and the real Bers image on the line

Let $\mathrm{Diff}_{\mathcal S}(\mathbb R)$ be the group of orientation-preserving diffeomorphisms $φ$ of the line with $φ'-1$ Schwartz and $φ(x)-x\to0$ as $x\to-\infty$, and let $β(φ)=\frac12S(φ)$ be half the Schwarzian derivative. We determine the image of $β$. The coordinate $w_φ=\frac12(\logφ')'$ identifies $\mathrm{Diff}_{\mathcal S}(\mathbb R)$ with the zero-mean hyperplane $\mathcal S_0(\mathbb R;\mathbb R)$ and turns $β$ into $w\mapsto w'-w^2$, so the question is which real Schwartz $q$ equal $w'-w^2$ for some $w$ of zero mean. Nonnegativity of the Schrodinger operator $H_q=-\partial_x^2-q$ decides which $q$ admit such a $w$ at all, and says nothing about the mean of $w$. The mean is a scattering invariant. Let $T_q$ and $R_q$ be the transmission and reflection coefficients of $H_q$. For every real $w\in\mathcal S(\mathbb R)$, with $q=w'-w^2$, we prove $T_q(0)=\mathrm{sech}(\int_{\mathbb R} w\,dx)$ and $R_q(0)=-\tanh(\int_{\mathbb R} w\,dx)$, so $\int_\mathbb R w\,dx$ is read off the scattering matrix at zero energy. Thus $q\inβ(\mathrm{Diff}_{\mathcal S}(\mathbb R))$ if and only if $H_q$ has no negative eigenvalues and $R_q(0)=0$, equivalently $T_q(0)=1$, and $φ\mapsto R_{β(φ)}$ is a bijection onto an explicit set of Schwartz reflection coefficients. We also compute the differential of $β$, whose range depends on the ambient topology. On the Schwartz space, that range is closed and split of codimension two, with normal functionals the first variations of the Wronskian of the two zero-energy Jost solutions and of $R_q(0)$. In every $W^{k,1}$ realization the second functional is unbounded. The range is then a dense proper subspace of the kernel of the first, hence neither closed nor split, and the operator has no bounded inverse on its range.

math.DG↗

arXiv2026

Intertwining the line bundle and Grauert-tube Hardy quantizations of the round 2-sphere

We compare two natural Hardy quantizations carried by the unit cosphere bundle of the round two-sphere. Through $S^2\simeq CP^1$, the cosphere bundle is the unit circle bundle of the canonical bundle $O(-2)$, and its Hardy space assembles the section spaces $H^0(CP^1,O(2\ell))$. Through the real-analytic round metric, the imaginary-time exponential map identifies the same cosphere bundle with every Grauert-tube boundary carrying the adapted complex structure, whose Reeb flow is the geodesic flow. Both Hardy spaces are realized in $L^2(SO(3))$, are multiplicity-free under the left action, and select one line in each Peter-Weyl multiplicity space. We compute the normalized overlap of these lines in closed form as $\frac{\sqrt{\binom{2\ell}{\ell}}}{2^\ell}\frac{\sinh^\ellτ}{\sqrt{P_\ell(\cosh 2τ)}}$, with $P_\ell$ the Legendre polynomial. The normalized kernel vectors define an explicit equivariant unitary between the two Hardy spaces, and the squared overlaps are the eigenvalues of the positive trace-class operator $Π_hΠ_τΠ_h$. We derive four exact trace series with the Hardy projectors inserted and show that their sum differs from the full flat trace by a distribution whose Abel regularization has cubic growth at every geodesic period. The eigenvalues also give a genus-zero Fredholm determinant of order zero, and truncating the complete large-$\ell$ expansion at any fixed order gives a finite polylogarithmic expression. A Bargmann-Fock calculation identifies the $\ell$-independent prefactor $(2\sinh 2τ)^{1/4}e^{-τ/2}$ in the large-$\ell$ asymptotic of the overlap with the Gaussian matrix coefficient of a metaplectic operator comparing the two contact planes.

math.SP↗

arXiv2026

Depth-Dependent Hidden-State Collapse in Dynamical System Autoencoders for LiDAR Point-Cloud Classification

We study Dynamical System Autoencoders (DSAE) for LiDAR point-cloud classification using spatial coordinates and Product Coefficient feature augmentations. The experiments compare separately trained DSAE architectures at encoder depths $K=1,\ldots,5$ and evaluate the resulting hidden representations with Random Forest, kNN, and a majority-class Dummy baseline. The main finding is a hidden-state collapse at $K=5$. For both xyz and xyz plus Product Coefficient inputs, the hidden-state standard deviation falls to the order of $10^{-5}$, while all three classifiers attain the same macro F1 score of $0.224688$. We prove that between-class hidden scatter is bounded by total hidden scatter, which in turn is controlled by the reported hidden-state variance. Thus a nearly constant hidden representation cannot retain substantial class-separating structure. Product Coefficients neither improve pre-collapse macro F1 nor prevent the $K=5$ collapse in the present DSAE setting. These results identify large-depth representation collapse as a concrete failure mode for DSAE LiDAR classification.

cs.LG↗

arXiv2026

Sampleability transport, nonlinear regularization, and the porous medium flow

We study the Wasserstein projection of a compactly supported probability measure onto the class of measures whose density ratio is bounded, and we place this projection in a broader program connecting generative modeling, optimal transport, and nonlinear diffusion. The paper proves existence and uniqueness of the sampleability projection, uniqueness of the Brenier map at the minimizer, path independence of the quadratic Wasserstein generation loss, and the diffusion-threshold picture for the heat semigroup. The porous medium equation is then analyzed as a candidate forward regularizer. We prove the two rigorous properties that make the equation attractive for this purpose, namely finite propagation of compact support and an explicit Wasserstein cost bound obtained from dissipation of the Rényi entropy. We then identify a structural obstruction inherent to any porous-medium version of the sampleability theory. Every nontrivial compactly supported whole-space porous-medium profile has vanishing essential infimum on any compact set containing its support, hence infinite density ratio in the original sense, and the assertion that the porous medium flow reaches the same density-ratio sampleable class while preserving compact support is false. To isolate the mathematically valid content of the nonlinear-diffusion program, we also prove an endpoint-constrained Benamou-Brenier principle for the sampleability projection and derive the corrected spectral picture near a strictly positive equilibrium on a fixed compact domain. In that regime the leading-order damping is exponential, with quadratic mode coupling in the first nonlinear correction. The Hele-Shaw and mesa-limit interpretation is therefore presented here as a conjectural variational extension rather than as a proved theorem.

math.AP↗