Determinantal PSD Graph Models

• The paper introduces a framework that fuses PSD matrix edge weights with a stabilized log-determinant energy to construct a determinantal graph Laplacian.
• It develops explicit first- and second-order differential formulas and a pullback Riemannian metric, allowing precise sensitivity and curvature analysis.
• The model underpins efficient cone-aware sampling methods and geometry-aware MALA, demonstrating superior performance in SPD matrix applications.

Searching arXiv for the cited papers to ground the article in the current literature. Determinantal PSD-weighted graph models are graph-based probabilistic and geometric models in which each edge of an undirected graph carries a positive semidefinite matrix weight, the global interaction operator is a block graph Laplacian, and the central scalar potential is a stabilized log-determinant energy of the form $\Phi(W)=-\log\det(L(W)+R)$ (Dey, 26 Mar 2026). In this formulation, the graph topology, the cone constraint on edge weights, and the determinant-based global coupling are fused into a single affine matrix map. The resulting framework supports explicit first- and second-order differential formulas, a pullback Riemannian metric on the PSD parameter space, sensitivity analysis via Hessian geometry, and cone-aware sampling procedures (Dey, 26 Mar 2026). It is closely related to the determinantal viewpoint of discrete determinantal point processes, where kernel matrices induce graphical Markov structure (Sadeghi et al., 2018), but it should not be conflated with kernel-quadratic “PSD models” for nonnegative function approximation, where “PSD” refers to positive semidefinite feature or kernel operators rather than determinantal graph structure (Marteau-Ferey et al., 2021).

1. Formal definition and parameter space

In the formulation studied in "Cone-Induced Geometry and Sampling for Determinantal PSD-Weighted Graph Models" (Dey, 26 Mar 2026), one starts with an undirected graph $G=(V,E)$ with $|V|=m$ and fixes a matrix size $d\ge 1$. Each edge $e\in E$ is assigned a PSD matrix weight

$W_e \in S_+^d,$

so the parameter space is the product cone

$K := (S_+^d)^E.$

After choosing an arbitrary orientation of $E$ and writing $B\in\mathbb R^{m\times |E|}$ for the oriented incidence matrix, the model defines the lifted block Laplacian

$$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$

This operator is independent of the chosen orientation and satisfies $G=(V,E)$0 for all $G=(V,E)$1 (Dey, 26 Mar 2026).

The associated quadratic form is

$G=(V,E)$2

for block vectors $G=(V,E)$3 with $G=(V,E)$4. This identifies the model’s native energy as a PSD-weighted Dirichlet energy on the graph (Dey, 26 Mar 2026). A fixed regularizer

$G=(V,E)$5

is then added, yielding

$G=(V,E)$6

This stabilized determinant construction is the defining mechanism of the determinantal PSD-weighted graph model (Dey, 26 Mar 2026).

A useful contextual distinction is that this object is neither a standard graph Laplacian model with scalar edge weights nor a generic kernel PSD density model. In the latter setting, a PSD model has the form

$G=(V,E)$7

or, in the Gaussian instantiation,

$G=(V,E)$8

which is a nonnegative function model on an input domain rather than a determinantal graph model (Marteau-Ferey et al., 2021). The shared use of “PSD” therefore refers to different mathematical roles.

2. Stabilized log-det energy and determinantal structure

The scalar energy of the model is

$G=(V,E)$9

with $|V|=m$0 ensuring that $|V|=m$1 is positive definite for all admissible $|V|=m$2 (Dey, 26 Mar 2026). The regularizer has two explicit functions in the construction: it ensures strict positivity and smoothness of the determinant and log-determinant, and it acts as a geometric regularizer preventing Laplacian degeneracy (Dey, 26 Mar 2026). The regularized energy is described as a stabilized log-det energy rather than a true barrier, since it stays finite near degeneracy and is correspondingly more numerically stable (Dey, 26 Mar 2026).

The determinantal aspect of the model lies in the use of $|V|=m$3 as the central scalar quantity. In the scalar case $|V|=m$4, the framework is linked to spanning-tree polynomials via matrix-tree-type theorems (Dey, 26 Mar 2026). This suggests a bridge between weighted graph combinatorics and cone-constrained matrix-valued edge interactions, although the stated result is the existence of the link rather than a full combinatorial reduction.

A separate determinantal tradition appears in discrete determinantal point processes. There, for a finite ground set $|V|=m$5, a DPP is defined by

$|V|=m$6

for a symmetric PSD kernel satisfying $|V|=m$7 (Sadeghi et al., 2018). In that setting, determinant identities encode probabilities of random subsets and give rise to graphical Markov properties. In the PSD-weighted graph model, by contrast, the determinant is applied to a stabilized block Laplacian generated by PSD edge parameters (Dey, 26 Mar 2026). Both frameworks are determinantal, but they organize different mathematical objects: kernel principal minors in the DPP case and a global Laplacian-plus-regularizer operator in the PSD-weighted graph case.

3. Differential calculus, convexity, and Rayleigh-type identities

Because $|V|=m$8 is affine, the derivatives of $|V|=m$9 are explicit (Dey, 26 Mar 2026). For cone directions $d\ge 1$0,

$d\ge 1$1

and

$d\ge 1$2

In particular,

$d\ge 1$3

so the log-det energy is convex along every cone direction (Dey, 26 Mar 2026).

The induced bilinear form

$d\ge 1$4

is the model’s pullback metric (Dey, 26 Mar 2026). For the determinant itself, the paper derives

$d\ge 1$5

$d\ge 1$6

and

$d\ge 1$7

These identities yield the Rayleigh-type factorization

$d\ge 1$8

equivalently,

$d\ge 1$9

Since $e\in E$0 for cone directions, the right-hand side is nonnegative; the paper identifies this as a continuous Rayleigh positivity certificate for the model (Dey, 26 Mar 2026).

These formulas give the framework an unusually explicit second-order structure. A plausible implication is that the model is especially amenable to curvature-based analysis because the Hessian can be evaluated without implicit differentiation or numerical linearization, but the explicit claim in the source is the availability of closed-form first and second derivatives and their use in geometry and sampling (Dey, 26 Mar 2026).

4. Pullback geometry on the product PSD cone

The geometry of the model is induced from the classical affine-invariant metric on the SPD cone,

$e\in E$1

which is also the Hessian of $e\in E$2:

$e\in E$3

Pulling this metric back along the affine map $e\in E$4 gives

$e\in E$5

on the PSD parameter space (Dey, 26 Mar 2026).

The associated local norm is

$e\in E$6

In the paper’s interpretation, this is a natural sensitivity geometry: it quantifies which PSD edge perturbations most strongly affect conditioning and determinant curvature (Dey, 26 Mar 2026). The same source also notes that because $e\in E$7 is self-concordant on $e\in E$8, the affine composition $e\in E$9 inherits self-concordance on any open set where $W_e \in S_+^d,$0, which provides controlled local variation of the Hessian metric and is useful for Newton-type methods and MALA discretizations (Dey, 26 Mar 2026).

This geometric construction has a natural relation to graphical interpretations of determinantal models, though the relation is indirect. In discrete DPPs, zeros in $W_e \in S_+^d,$1 define a bidirected marginal-independence graph and zeros in $W_e \in S_+^d,$2 define an undirected graph for context-specific conditioning events such as $W_e \in S_+^d,$3 (Sadeghi et al., 2018). In determinantal PSD-weighted graph models, by contrast, the graph $W_e \in S_+^d,$4 is given a priori and the determinant is built from its block Laplacian rather than from principal minors of a DPP kernel (Dey, 26 Mar 2026). The shared determinant machinery thus underwrites different notions of structure: conditional independence in one case, cone-induced sensitivity geometry in the other.

5. Sensitivity analysis and low-dimensional validation

The low-dimensional validation in (Dey, 26 Mar 2026) concentrates especially on $W_e \in S_+^d,$5 and uses rank-one PSD perturbations. For an edge $W_e \in S_+^d,$6 and vector $W_e \in S_+^d,$7, the perturbation supported only on edge $W_e \in S_+^d,$8 is defined by

$W_e \in S_+^d,$9

Its lifted operator perturbation is

$K := (S_+^d)^E.$0

The intrinsic curvature score attached to this perturbation is

$K := (S_+^d)^E.$1

which measures local sensitivity to that PSD edge direction (Dey, 26 Mar 2026).

The paper then compares the exact metric score with a finite-difference curvature approximation

$K := (S_+^d)^E.$2

where

$K := (S_+^d)^E.$3

The reported finding is that $K := (S_+^d)^E.$4 tracks $K := (S_+^d)^E.$5 closely, confirming numerical accuracy of the Hessian metric in this regime (Dey, 26 Mar 2026).

For collections of perturbations $K := (S_+^d)^E.$6, the paper ranks them by the score

$K := (S_+^d)^E.$7

and defines the capture curve

$K := (S_+^d)^E.$8

Ranking by the Hessian score yields near-oracle recovery of total sensitivity mass and clearly outperforms random ranking (Dey, 26 Mar 2026). This supports the interpretation of the pullback metric as an operational tool for identifying locally important perturbation directions, especially among rank-one PSD edge perturbations.

A plausible implication is that the geometry may be useful for adaptive model reduction or targeted edge-direction exploration, but the explicit result stated in the source is sensitivity ranking rather than a reduction algorithm (Dey, 26 Mar 2026).

6. Intrinsic Gibbs laws and geometry-aware sampling

The same geometry is used to define sampling schemes on cone-supported spaces (Dey, 26 Mar 2026). On a cone interior or cone-based manifold $K := (S_+^d)^E.$9, the framework introduces

$E$0

and when $E$1 is strictly concave, $E$2 is positive definite and defines a Hessian Riemannian metric

$E$3

The induced volume form is

$E$4

Given an external energy $E$5, the intrinsic Gibbs law is

$E$6

The paper then invokes the Bakry–Émery criterion: if

$E$7

then $E$8 satisfies a Poincaré inequality and $E$9, giving Gaussian concentration for Lipschitz observables and entropy decay for the overdamped Langevin diffusion

$B\in\mathbb R^{m\times |E|}$0

It also yields

$B\in\mathbb R^{m\times |E|}$1

and the paper further invokes Talagrand’s $B\in\mathbb R^{m\times |E|}$2 inequality to bound $B\in\mathbb R^{m\times |E|}$3-distance and observable bias (Dey, 26 Mar 2026).

For the SPD cone $B\in\mathbb R^{m\times |E|}$4 equipped with the affine-invariant metric

$B\in\mathbb R^{m\times |E|}$5

the paper gives an explicit geometry-aware Metropolis-adjusted Langevin algorithm. The Riemannian gradient is

$B\in\mathbb R^{m\times |E|}$6

Using congruence coordinates $B\in\mathbb R^{m\times |E|}$7, the exponential map is

$B\in\mathbb R^{m\times |E|}$8

The Langevin increment in tangent coordinates is

$B\in\mathbb R^{m\times |E|}$9

and the proposal becomes

$$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$0

Because the proposal density is taken with respect to $$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$1, the exponential-map Jacobian must be included:

$$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$2

in general, and on $$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$3,

$$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$4

where $$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$5 are the eigenvalues of $$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$6. The Metropolis–Hastings acceptance rule is

$$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$7

(Dey, 26 Mar 2026).

In the SPD $$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$8 example, the final computational comparison is between geom_MALA and naive_Euclid_drift_in_S. Reported diagnostics include acceptance, split-$$L(W) := (B\otimes I_d)\left(\bigoplus_{e\in E} W_e\right)(B^\top\otimes I_d).$$9, ESS/sec, ECDF overlays, pooled histograms, and an empirical Poincaré proxy (Dey, 26 Mar 2026). The geometry-aware sampler attains acceptance

$G=(V,E)$00

has substantially larger ESS/sec on observables such as $G=(V,E)$01, $G=(V,E)$02, and $G=(V,E)$03, and yields $G=(V,E)$04 values essentially equal to 1 for both methods (Dey, 26 Mar 2026). ECDF and histogram overlays show that both samplers target the same distribution, while the geometry-aware chain is much more efficient (Dey, 26 Mar 2026). The empirical Poincaré proxy remains of similar magnitude across methods, supporting consistency of the target law under both proposals (Dey, 26 Mar 2026).

7. Relations, distinctions, and conceptual scope

Determinantal PSD-weighted graph models sit at the intersection of graph Laplacians, cone-constrained matrix analysis, log-det geometry, and determinantal modeling (Dey, 26 Mar 2026). Their determinant is global and operator-based, and their geometry is generated by the Hessian of a stabilized log-det energy. This is distinct from two nearby literatures.

First, they differ from discrete DPP graphical models. In DPPs, the kernel $G=(V,E)$05 itself determines graphical independences. Specifically,

$G=(V,E)$06

so zeros in $G=(V,E)$07 define a bidirected marginal-independence graph, while

$G=(V,E)$08

which gives an undirected graphical interpretation for context-specific conditional independence (Sadeghi et al., 2018). The paper further states that DPPs are faithful to their induced bidirected or undirected graphs only under additional stability conditions, so Markovness does not automatically imply faithfulness (Sadeghi et al., 2018). In determinantal PSD-weighted graph models, however, the graph is not inferred from zero patterns of a kernel; it is part of the parameterization, and the determinant defines energy and geometry rather than subset probabilities (Dey, 26 Mar 2026).

Second, they differ from nonnegative-function PSD models. In "Sampling from Arbitrary Functions via PSD Models" (Marteau-Ferey et al., 2021), a PSD model means a nonnegative function represented as a quadratic form in features,

$G=(V,E)$09

or, for Gaussian PSD models,

$G=(V,E)$10

with exact hyper-rectangle integration and a recursive dyadic sampler for approximate i.i.d. sampling from the fitted model (Marteau-Ferey et al., 2021). That framework is explicitly stated not to discuss determinantal point processes, determinant-based sampling, or weighted graph models; the matrix $G=(V,E)$11 is a coefficient matrix in a kernel expansion, not an adjacency or graph weight matrix (Marteau-Ferey et al., 2021). This distinction is important because the shared acronym “PSD” can obscure fundamentally different uses: positive semidefinite coefficients in function approximation versus positive semidefinite edge weights in a determinantal graph operator.

Within its own scope, the determinantal PSD-weighted graph model provides a computational pipeline from PSD-weighted graph specification to explicit derivatives, induced Riemannian geometry, sensitivity ranking, and cone-aware Monte Carlo methods (Dey, 26 Mar 2026). This suggests a unifying perspective in which determinant-based graph energies are not only objectives but also generators of intrinsic geometry and sampling dynamics.