pith. sign in

arxiv: 2305.02635 · v2 · pith:BQXVKTPTnew · submitted 2023-05-04 · 🧮 math.FA

Deconvolution on graphs via linear programming

Bernhard G. Bodmann , Jennifer J. May This is my paper

Pith reviewed 2026-05-24 08:39 UTC · model grok-4.3

classification 🧮 math.FA
keywords deconvolutionheat kernelsgraphslinear programmingheat semigroupsignal recoveryfunctional analysis
0
0 comments X

The pith

A linear program recovers the individual heat kernels in a superposition on a graph when the centers satisfy geometric separation conditions and the time parameter is sufficiently small.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to separate the components of a sum of heat kernels centered at different vertices on a graph by solving a linear program. It derives geometric conditions on how far apart those vertices must be and upper bounds on the time parameter of the heat semigroup that make the recovery unique and successful. The result is presented first for a shared time parameter across all kernels and then for the case where each kernel evolves for a time that depends on its center. A reader interested in disentangling mixed signals on networks would see this as a concrete method for performing deconvolution on discrete structures.

Core claim

Under geometric conditions on the vertices at which heat kernels are centered and bounds on the time parameter governing the evolution under the heat semigroup, a linear program guarantees successful recovery of the individual terms in their superposition. The result holds for both a common time parameter and when the time parameter depends on the location.

What carries the argument

The linear program that recovers the coefficients of the heat-kernel superposition from observed data on the graph.

If this is right

  • The locations and amplitudes of the individual heat kernels can be uniquely recovered from the observed sum.
  • Recovery succeeds when every kernel uses the same time parameter.
  • Recovery also succeeds when each kernel is allowed its own location-dependent time parameter.
  • The method applies directly to any graph whose vertices meet the required separation criteria.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same linear-programming strategy might apply to superpositions of other kernels generated by the graph Laplacian.
  • The separation conditions could be checked algorithmically before attempting recovery on a concrete network.
  • The two time-parameter settings indicate that the recovery procedure tolerates moderate variation in diffusion times across sources.

Load-bearing premise

The vertices where the heat kernels are centered must satisfy the stated geometric separation conditions; without them the linear program may not distinguish the individual terms.

What would settle it

A graph whose vertices violate the separation conditions, together with explicit coefficients, such that solving the linear program yields a unique but incorrect decomposition of the observed superposition.

read the original abstract

The main challenge addressed in this paper is to identify individual terms in a superposition of heat kernels on a graph. We establish geometric conditions on the vertices at which these heat kernels are centered and find bounds on the time parameter governing the evolution under the heat semigroup that guarantee a successful recovery. This result can be viewed as a type of deconvolution on a graph. A first main result addresses the setting of a common time parameter for all the heat kernels. We also treat a more general setting when the time parameter depends on the location at which the heat kernel is centered.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript develops a linear programming framework for recovering the individual components of a superposition of heat kernels on a graph. It establishes geometric conditions on the vertices at which the kernels are centered together with bounds on the time parameter (both in the uniform-time case and the location-dependent-time case) that guarantee exact recovery of the superposition via the LP.

Significance. If the stated geometric and temporal conditions suffice for recovery, the work supplies a theoretically grounded deconvolution procedure on graphs that is computationally tractable and rests on verifiable separation assumptions rather than on fitted parameters. This constitutes a concrete contribution to graph signal processing and to the analysis of the heat semigroup on discrete structures.

minor comments (3)
  1. The abstract and introduction would benefit from an explicit statement of the precise linear program that is solved (variables, objective, and constraints) so that the geometric conditions can be read against the optimization problem without first consulting later sections.
  2. Notation for the heat kernel and the heat semigroup should be introduced once and used consistently; occasional shifts between K_t(x,·) and p_t(x,·) are distracting.
  3. A short remark on how the geometric separation condition can be checked algorithmically on a given graph would help readers assess practical applicability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes geometric conditions on vertex centers and bounds on the time parameter of the heat semigroup to guarantee that a linear program recovers individual heat kernel terms in a superposition. This is framed as a theorem deriving sufficient conditions for deconvolution on graphs, with no fitted parameters renamed as predictions, no self-definitional loops in the equations, and no load-bearing self-citations or ansatzes imported from prior work by the same authors. The result is presented as following from the properties of the heat semigroup and linear programming duality on graphs, remaining independent of its own inputs. The provided abstract and claim description contain no steps that reduce by construction to the target recovery statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the work rests on standard properties of the heat semigroup on graphs from functional analysis; no free parameters, invented entities, or ad-hoc axioms are visible.

axioms (1)
  • standard math The heat semigroup on the graph is well-defined and satisfies standard diffusion properties from spectral graph theory.
    Invoked implicitly when referring to evolution under the heat semigroup.

pith-pipeline@v0.9.0 · 5610 in / 1111 out tokens · 19520 ms · 2026-05-24T08:39:24.895301+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

  1. [1]

    B. G. Bodmann, A. Flinth, and G. Kutyniok. Compressed sen sing for analog signals, https://arxiv.org/abs/1803.042 18, 2018

    work page 2018

  2. [2]

    Cand` es, J

    E. Cand` es, J. Romberg, and T. Tao. Stable signal recover y from incomplete and inaccurate measurements. Commun. Pure Appl. Math. , pages 1207–1223, 2005

    work page 2005

  3. [3]

    Cand` es, M

    E. Cand` es, M. Rudelson, T. Tao, and R. Vershynin. Error c orrection via linear programming. In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05) , pages 668–681. IEEE, 2005

    work page 2005

  4. [4]

    E. J. Cand` es and C. Fernandez-Granda. Towards a mathema tical theory of super-resolution. Communications on Pure and Applied Mathematics , 67(6):906–956, 2014

    work page 2014

  5. [5]

    E. J. Cand` es and Y. Plan. A probabilistic and ripless the ory of compressed sensing. IEEE transactions on Information Theory, 57(11):7235–7254, 2011

    work page 2011

  6. [6]

    L. F. O. Chamon and A. Ribeiro. Greedy sampling of graph si gnals. IEEE Trans. Signal Process. , 66(1):34–47, 2018

    work page 2018

  7. [7]

    S. Chen, A. Sandryhaila, J. M. F. Moura, and J. Kovaˇ cevi´ c. Signal recovery on graphs: variation minimization. IEEE Trans. Signal Process., 63(17):4609–4624, 2015

    work page 2015

  8. [8]

    E. B. Davies. Large Deviations for Heat Kernels on Graphs . Journal of the London Mathematical Society , s2-47(1):65–72, 02 1993

    work page 1993

  9. [9]

    E.B. Davies. Analysis on graphs and noncommutative geom etry. Journal of functional analysis , 111(2):398–430, 1993

    work page 1993

  10. [10]

    M. Folz. Gaussian upper bounds for heat kernels of conti nuous time simple random walks. Elec. J. Prob. , 62:1693–1722, 2011

    work page 2011

  11. [11]

    Foucart and H

    S. Foucart and H. Rauhut. A Mathematical Introduction to Compressive Sensing . Birkhauser, Boston, 2013

    work page 2013

  12. [12]

    R. A. Horn and C. R. Johnson. Matrix analysis . Cambridge University Press, Cambridge [Cambridgeshire] ;, 1985

    work page 1985

  13. [13]

    I. Z. Pesenson and M. Z. Pesenson. Sampling, filtering an d sparse approximations on combinatorial graphs. J. Fourier Anal. Appl. , 16(6):921–942, 2010

    work page 2010

  14. [14]

    Sandryhaila and J

    A. Sandryhaila and J. M. F. Moura. Discrete signal proce ssing on graphs. IEEE Trans. Signal Process. , 61(7):1644–1656, 2013

    work page 2013

  15. [15]

    Sandryhaila and J

    A. Sandryhaila and J. M. F. Moura. Discrete signal proce ssing on graphs: frequency analysis. IEEE Trans. Signal Process., 62(12):3042–3054, 2014

    work page 2014

  16. [16]

    Y. Tanaka. Spectral domain sampling of graph signals. IEEE Trans. Signal Process. , 66(14):3752–3767, 2018

    work page 2018

  17. [17]

    G. Tang, B. N. Bhaskar, P. Shah, and B. Recht. Compressed sensing off the grid. IEEE transactions on information theory , 59(11):7465–7490, 2013

    work page 2013

  18. [18]

    M. Unser. A unifying representer theorem for inverse pr oblems and machine learning. Foundations of Computational Mathematics, 21(4):941–960, 2021

    work page 2021

  19. [19]

    J.M. Varah. A lower bound for the smallest singular valu e of a matrix. Linear Algebra and its Applications , 11(1):3–5, 1975. 641 Philip G. Hoffman Hall, Department of Mathematics, Univers ity of Houston, Houston, TX 77204-3008 Email address : bgb@math.uh.edu

    work page 1975