arXiv subjects

James Tian

Publications and source records attributed to James Tian.

At least 19 recordsLinked to original sources

arXiv2026

The role of parameter Jacobians in the stability of network outputs

In the framework of network dynamics, learning models, and neural tangent kernels (NTK), we show that the corresponding linearized dynamics leads naturally to a semigroup formulation. More precisely, in our analysis of input/output models, the time-dynamics is presented via special semigroups of linear operators on Hilbert spaces, together with an associated class of semigroup perturbations. In this context, we then present new and explicit a priori perturbation-bound results: for the fixed-kernel linearization constructions arising in the NTK setting, we prove norm-bounds on the corresponding semigroup perturbations, in the form of explicit finite-time perturbation estimates. We further present refinements on prescribed task spaces, Ces\`aro-averaged (ergodic) comparisons estimates, and versions in which the lower spectral edge assumption is replaced by a spectral-distribution condition. We also extend the comparison to nonautonomous NTK evolutions through piecewise-frozen approximations, record a corresponding discrete Euler specialization, and offer worked examples in order to illustrate our perturbation-bound estimates.

math.FA↗

arXiv2026

A finite-Point Schwarz-Pick Inequality

We prove a finite-point version of the Schwarz-Pick inequality for holomorphic self-maps of the disk. The estimate compares Bergman Gram matrices built from several points and their images, and the sharp constant depends on both the number of points and, for finite Blaschke products, the degree of the map. These matrices also arise from the Szeg\H{o}-kernel metric on the symmetrized polydisc, where the estimate becomes a sharp Schwarz-Pick inequality for the induced holomorphic maps.

math.CV↗

arXiv2026

Tree Coordinates and Range Martingales for Positive Operator-Valued Measures

Positive operator-valued measures on a tree admit intrinsic local coordinates coming from the way each cylinder value splits into its children. We show that these local splittings, taken on the range spaces of the cylinder values, recover the measure and at the same time build an intrinsic direct limit dilation whose cylinder projections yield the minimal Naimark dilation. In these coordinates, the commutant of the dilation becomes a martingale calculus on the range spaces. This gives local descriptions of extremality and domination, and it also yields a bounded change-of-measure transform that updates the tree coordinates in a natural way. For self-adjoint range martingales we obtain a quadratic variation formula from the range space isometries, and the associated local variance terms detect the projection-valued case.

math.PR↗

arXiv2026

Compression Covariance and Tangent kernels

Let $A\geq0$ be self-adjoint on a Hilbert space $H$, let $T_{t}=e^{-tA}$, and let $P$ be an orthogonal projection. Relative to the decomposition $H=PH\oplus P^{\perp}H$, write \[ T_{t}=\begin{pmatrix}C_{t} & V^{*}_{t}\\ V_{t} & D_{t} \end{pmatrix}, \] where $C_{t}=PT_{t}P|_{PH}$, $V_{t}=P^{\perp}T_{t}P|_{PH}$, and $D_{t}=P^{\perp}T_{t}P^{\perp}|_{P^{\perp}H}$. The compressed family $\left(C_{t}\right)$ consists of positive contractions but need not form a semigroup. Its defect is given by \[ C_{s+t}-C_{s}C_{t}=V^{*}_{s}V_{t} \] while the complementary block satisfies \[ D_{s+t}-D_{s}D_{t}=V_{s}V^{*}_{t}. \] Thus the failure of $\left\{ C_{t}\right\} $ and $\left\{ D_{t}\right\} $ to be semigroups gives two Gram kernels associated with the same off-diagonal maps. We treat these covariance defects as positive definite operator-valued kernels and use their Kolmogorov spaces to recover the hidden dynamics they encode. We then study short-time rescalings of $E_{s,t}:=V^{*}_{s}V_{t}$. The tangent kernel \[ F\left(s,t\right):=\lim_{\varepsilon\downarrow0}a\left(\varepsilon\right)^{-1}E_{\varepsilon s,\varepsilon t} \] has its own Kolmogorov space, and the lower-right block dynamics induces a positive self-adjoint contraction semigroup on it. The representing vectors of $F$ then satisfy an additive cocycle identity for this semigroup. This gives an intrinsic restriction on the positive kernels that can arise as short-time compression covariance tangents.

math.FA↗

arXiv2026

Ordered POVMs and Residual Collapse

Ordered realizations of discrete POVMs are studied through a residual transform generated by sequential tests. One application of the transform replaces each coordinate by the effect obtained after all earlier tests have failed, and appends the remaining mass as a terminal outcome. Under natural hypotheses, iterating the transform produces a collapsed POVM whose non-escape coordinates are the parts of the original effects that survive all earlier tests. The resulting collapse map gives an equivalence relation on ordered POVM realizations. Its range and fibers are characterized. The range consists of collapsed POVMs, whose non-escape coordinates are mutually orthogonal and whose support projections strongly sum to the identity. The fiber over a collapsed POVM consists of all ordered realizations with the same residually visible compressions. In particular, different ordered realizations, including ones with different off-diagonal coupling data, can have the same collapsed image. After collapse, the non-escape coordinates are fixed under further residual iteration. The remaining dynamics takes place in the escape effect, which is fragmented by a universal scalar functional calculus.

math.FA↗

arXiv2026

Bernoulli cylinder frame operators: filtration, Haar structure, and self-similarity

We study the finite-rank frame operators generated by cylinder indicator functions for the Bernoulli Cantor measure $\mu_{p}$. In the symmetric case $p=\frac{1}{2}$, the natural Haar differences diagonalize these operators. For general $0<p<1$, we show that the weighted Haar basis still yields a sparse tree-banded matrix form, although diagonalization is lost. We also prove a filtration representation in terms of conditional expectations and level-wise mass operators. This leads to a norm convergent limit operator $K_{\infty}$, which is compact, positive, and self-adjoint. Finally, we show that $K_{\infty}$ is characterized by a self-similar operator identity induced by the first-level Cantor decomposition, and we derive corresponding block and scalar resolvent renormalization formulas.

math.FA↗

arXiv2026

Classification in Active Dimension 2 for Weighted Residual Dynamics

We study weighted residual dynamics associated with a rank-one projection in finite dimension. The iteration reduces, after finitely many steps, to a nonlinear recursion on a stabilized active subspace. We prove that this recursion can be classified when the active dimension is two: either a transverse reducing direction persists unchanged, or the coupled part collapses completely. As a consequence, we obtain a description of the limit in the active two-dimensional case and identify the threshold beyond which higher-dimensional behavior becomes more flexible.

math.FA↗

arXiv2026

Multipartite parity bounds and total correlation

This paper studies multipartite observables formed from sums of local self-adjoint contractions on tensor product Hilbert spaces. The square of such a sum has a parity structure: after decomposing each local product into commutator and anticommutator parts, the odd parity terms cancel and only even parity contributions remain. This yields a norm bound in terms of a family of pairwise defect weights built from local commutator and anticommutator norms. These defect weights also control an information theoretic estimate. The excess of the observable expectation above the product state threshold is shown to necessarily carry a definite amount of total correlation. Under a natural $\ell^{2}$-type bound on each local family, this product state threshold becomes explicit, which leads to a fully explicit lower bound on total correlation. A simple depolarizing example illustrates the resulting decay mechanism under local noise.

quant-ph↗

arXiv2026

Tree Capacity and Splitting Isometries for Subinvariant Kernels

Starting from a subinvariant positive definite kernel under a branching pullback, we attach to the resulting kernel tower a canonical electrical network on the word tree whose edge weights are the diagonal increments. This converts diagonal growth into effective resistance and capacity, giving explicit criteria and quantitative bounds, together with a matching upper bound under a mild level regularity condition. When the diagonal tower has finite limit at a point, we prove convergence of the full kernels and obtain an invariant completion with a minimality property. We also describe the associated RKHS splitting and a boundary martingale construction leading to weighted invariant majorants.

math.PR↗

arXiv2026

Subinvariant kernel dynamics

We study positive definite kernels pulled back along a finite family of self-maps under a subinvariance inequality for the associated branching operator. Iteration produces an increasing kernel tower with defect kernels. Under diagonal boundedness, the tower has a smallest invariant majorant, with a canonical defect space realization and an explicit diagonal harmonic envelope governing finiteness versus blow-up. We also give probabilistic and boundary representations: a Gaussian martingale model whose quadratic variation is the defect sequence, and canonical Doob path measures with a boundary feature model for the normalized defects.

math.PR↗

arXiv2026

Boundary Disintegration for Weighted Residual Energy Trees

We study iterated weighted residual (WR) splittings generated by a positive operator $R_{0}\in B\left(H\right)_{+}$ and a finite family of contractions $C_{1},\dots,C_{m}$ in $B\left(H\right)$. The associated residual update $R\mapsto R^{1/2}(I-C^{*}_{j}C_{j})R^{1/2}$ produces an $m$-ary energy tree of residuals $\left\{ R_{w}\right\} $ and dissipated pieces $\left\{ D_{w,j}\right\} $ indexed by finite words. From this tree we construct intrinsic path measures on the path space by biasing transitions either by a fixed quadratic form $x\mapsto\left\langle x,D_{w,j}x\right\rangle $ (defining the measures $\nu_{x}$) or, in the trace-class setting, by ${\rm tr}\left(D_{w,j}\right)$ (yielding a reference measure $\nu_{\mathrm{tr}}$). When $R_{0}\in S_{1}\left(H\right)_{+}$, we show that $\nu_{\mathrm{tr}}$ dominates the family $\left\{ \nu_{x}\right\} $ and identify $d\nu_{x}/d\nu_{\mathrm{tr}}$ as a canonical martingale limit of cylinder likelihood ratios. Along $\nu_{\mathrm{tr}}$-almost every branch the residuals decrease to a terminal trace-class random variable $R_{\infty}$, which we interpret as the WR boundary variable. We then disintegrate $\nu_{\mathrm{tr}}$ over $\sigma\left(R_{\infty}\right)$, obtaining a boundary law $\mu_{\mathrm{tr}}=\left(R_{\infty}\right)_{\#}\nu_{\mathrm{tr}}$ and conditional path measures $\left\{ \nu^{T}_{\mathrm{tr}}\right\} $. Finally, we show that each $\nu_{x}$ admits a boundary representation as a mixture of $\left\{ \nu^{T}_{\mathrm{tr}}\right\} $ with an explicit boundary density $h_{x}=d\mu_{x}/d\mu_{\mathrm{tr}}$, thereby organizing the family of intrinsic WR path measures by a single trace-biased boundary disintegration.

math.PR↗

arXiv2026

Wavelet-Packet Content for Positive Operators

We study positive operator decompositions associated with rooted trees of orthogonal projections. In this sense, the refinement tree induces an ``MRA in $B\left(H\right)_{+}$''. To each node we assign a positive content operator, and these contents split along the tree and yield a positive decomposition at each fixed depth. The resulting decomposition gives a multiresolution description of positive operators adapted to the tree. In the trace class setting, the scalar contents determine a canonical boundary measure on the path space, and for each vector the corresponding quadratic data admit a nonnegative integrable density with respect to that measure. At fixed depth, we study greedy extraction rules based on trace and Hilbert-Schmidt norm. The trace rule gives a sharp geometric decay estimate for the trace of the positive remainder. In the Hilbert-Schmidt setting, a depth dependent coherence parameter measures departure from block diagonal form and yields geometric decay bounds. We also study adaptive partitions up to a terminal depth. In that setting, the change in total squared content under local refinement is determined by off-diagonal interaction among the child contents. This leads to an additive refinement calculus for adaptive decompositions and recursive criteria for optimal adaptive partitions.

math.FA↗

arXiv2026

Random Frame Decompositions from Weighted Residual Flows

We study the evolution of a positive operator under weighted residual maps determined by a finite family of orthogonal projections. Iterating these maps along the rooted tree of multi-indices produces a "weighted residual energy tree", together with natural path measures obtained by normalizing the dissipated energy or trace at each step. Under a quantitative coverage condition on the projections, we show that along almost every branch the residuals converge strongly to zero and the dissipated pieces admit a rank-one decomposition that reconstructs the initial operator. In the special case where the initial operator is the identity on a subspace, this yields almost surely a random Parseval frame generated intrinsically by the weighted residual dynamics.

math.FA↗

arXiv2025

Alternating Weighted Residual Flows and the Non-Commutative Gap

This work develops a nonlinear analogue of alternating projections on Hilbert space, based on iterating a weighted residual transformation that removes the portion of an operator detected by a projection after conjugation by its square root. Although this map is neither linear nor variational and falls outside classical operator-mean frameworks, the alternating flow between two fixed projections is shown to be monotone and to converge strongly to a positive limit supported on their common kernel. The analysis identifies an intrinsic representation of this limit inside the operator range of the initial datum, which makes it possible to compare the nonlinear limit with the shorted operator of Anderson-Duffin-Trapp. The nonlinear flow always produces an operator dominated by the shorted operator, with equality precisely in the commuting regime. A global energy identity describes how mass is dissipated at each step of the iteration, and a factorized description localizes the gap between the nonlinear limit and the classical shorted operator.

math.FA↗

arXiv2025

Shorting Dynamics and Structured Kernel Regularization

This paper develops a nonlinear operator dynamic that progressively removes the influence of a prescribed feature subspace while retaining maximal structure elsewhere. The induced sequence of positive operators is monotone, admits an exact residual decomposition, and converges to the classical shorted operator. Transporting this dynamic to reproducing kernel Hilbert spaces yields a corresponding family of kernels that converges to the largest kernel dominated by the original one and annihilating the given subspace. In the finite-sample setting, the associated Gram operators inherit a structured residual decomposition that leads to a canonical form of kernel ridge regression and a principled way to enforce nuisance invariance. This gives a unified operator-analytic approach to invariant kernel construction and structured regularization in data analysis.

math.FA↗

arXiv2025

A Kernel Approach to the Stinespring and Kraus Representations

We give a self-contained derivation of the Stinespring theorem for completely positive maps and the Kraus representation for normal completely positive maps using two scalar positive definite kernels. The first kernel realizes the minimal Stinespring space directly as a reproducing kernel Hilbert space, without passing through a quotient construction, while the second separates the multiplicity space and produces the Kraus operators as its coordinate operators. For holomorphic self-maps $f$ of the disk, this identifies the de Branges-Rovnyak space $\mathcal{H}\left(f\right)$ as the canonical Kraus multiplicity space of the associated map $\Phi_{f}$ on $B\left(H^{2}\right)$. Composition of symbols is reflected by canonical isometries between these multiplicity spaces, and the same kernel formulas describe the iterates of $\Phi_{f}$. We show in particular that every $\Phi_{f}$ is extreme and that, when the iterates of $f$ converge to a point of the disk, the preadjoint iterates converge in trace norm to the corresponding Szeg\H{o} kernel state.

math.FA↗

arXiv2025

Residual-Weighted Decomposition of Positive Operators

This paper investigates an iterative rank-one decomposition scheme for positive operators on a Hilbert space based on a residual-weighted congruence update. At each step the operator is compressed along a chosen unit vector while remaining inside the positive cone, and the resulting map defines a monotone dynamical system on the cone of positive operators. We prove that the associated residuals admit a canonical telescoping decomposition into rank-one terms and a limiting positive operator, and we identify this limit together with an exact energy identity expressing the defect between the initial and limiting operators as a convergent series of rank-one contributions. In the case where the iteration exhausts the operator, the residual directions form a Parseval frame for the natural range space, yielding a constructive procedure that produces Parseval frames without spectral calculus. We further solve the inverse problem by characterizing those decreasing chains with rank-one steps that arise from such dynamics via an intrinsic normalization condition involving the Moore-Penrose inverse. For trace-class operators we obtain a scalar energy identity and show that mild greedy or density conditions on the chosen directions guarantee exhaustion. An application to reproducing kernel Hilbert spaces illustrates the abstract results.

math.FA↗

arXiv2025

Universal Kernel Models for Iterated Completely Positive Maps

We study how iterated and composed completely positive maps act on operator-valued kernels. Each kernel is realized inside a single Hilbert space where composition corresponds to applying bounded creation operators to feature vectors. This model yields a direct formula for every iterated kernel and allows pointwise limits, contractive behavior, and kernel domination to be read as standard operator facts. The main results include an explicit limit kernel for unital maps, a Stein-type decomposition, a Radon-Nikodym representation under subunitality, and an almost-sure growth law for random compositions. The construction keeps all iterates in one space, making their comparison and asymptotic analysis transparent.

math.FA↗