Pith. sign in

REVIEW 3 major objections 4 minor 39 references

A Class of Doubly Stochastic Shift Operators for Random Graph Signals and their Boundedness

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proposes a class of doubly stochastic graph shift operators and shows that, for locally stationary random graph signals, they are asymptotically $L_2$-bounded, and asymptotically $L_2$-isometric for i.i.d.

desk verdict The paper's headline asymptotic consistency result for doubly stochastic GSOs is false as stated; fixed-N bounds are fine, but the limit argument rests on an impossible invariance assumption and the L2-isometry label is wrong. read the letter →

arxiv 1908.01596 v5 pith:2BXV7IR2 submitted 2019-08-05 eess.SP cs.ITmath.IT

Bruno Scalzo Dees , Ljubisa Stankovic , Milos Dakovic , Anthony G. Constantinides , Danilo P. Mandic This is my paper · ORCID
classification eess.SPcs.ITmath.IT
keywords graphsignalprocessingdoublystochasticmatrixshiftoperatorstatisticalconsistencyL2boundednessisometrylocallystationarysignalsKantorovichinequality
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes using doubly stochastic matrices---nonnegative matrices whose rows and columns each sum to one---as graph shift operators, and asks what happens to the energy of a random graph signal when such a shift is applied. It establishes three properties: the shift preserves the signal mean exactly; for locally stationary random signals it is upper and lower bounded in expected squared amplitude, with the bound governed by the signal's correlation $\rho$ and the ratio of the largest to smallest shift entries; and for i.i.d. signals it becomes an $L_2$-isometry in the limit of large incoming neighborhoods, meaning the expected power of the shifted signal converges to the square of the mean. The interest is that ordinary adjacency or Laplacian shifts can distort signal spectra, while this class gives bounded, nearly isometric shifts and comes with a practical averaging and denoising interpretation.

What carries the argument

The central object is the doubly stochastic graph shift operator $S\in\mathbb{R}^{N\times N}$, defined by $S_{mn}\ge 0$, $S\mathbf{1}=\mathbf{1}$, $S^T\mathbf{1}=\mathbf{1}$. Its left-stochastic half gives a Markov diffusion interpretation, since each column is a set of transition probabilities of a random walker; its right-stochastic half makes each output an unbiased expectation operator, since each row sums to one. The quantitative engine is the Kantorovich inequality applied to the squared row entries, which converts the bounds $L$ and $U$ on shift entries into the factor $(L+U)^2/(4LU)$; the AM-GM inequality then identifies this factor as the squared ratio of the arithmetic to the geometric mean of $L$ and $U$. This machinery reduces boundedness of the shift to a statistical consistency analysis of a graph-dependent average.

What would settle it

Take a family of graphs with increasing incoming neighborhood size $N_m$, such as directed stars with edge weights that decay along the leaves, compute the doubly stochastic normalization, and track $\sum_n S_{mn}^2$ together with the factor $(L+U)^2/(4LU)$; if $N_m\sum_n S_{mn}^2$ does not stay bounded or $L$ tends to $0$ while $U$ does not, then the claimed limiting bound in eq. (21) fails for that family.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the doubly stochastic property turns a graph shift into a statistically consistent estimator: each output $S(x_m)=\sum_{n\in V_m} S_{mn}x_n$ is an unbiased estimate of the local mean $\mu$, and its variance can be controlled. Using the Kantorovich inequality, the paper shows that $\sum_{n\in V_m} S_{mn}^2 \le \frac{1}{N_m}\frac{(L+U)^2}{4LU}$, with $0<L\le S_{mn}\le U<1$, so the variance of the shift satisfies $\lim_{N_m\to\infty} \operatorname{var}\{S(x_m)\} \le \rho\sigma^2 \frac{(L+U)^2}{4LU}$. For i.i.d. signals ($\rho=0$) the variance vanishes and $\lim_{N_m\to\infty} E\{S(x_m)^2\}=\mu^2$, which the paper calls asymptotic $L_2$-isometry; in general the shift is asymptotically $L_2$-bounded with a bias term equal to the squared arithmetic-to-geometric mean ratio of $L$ and $U$.

Load-bearing premise

The proof that the variance vanishes for i.i.d. signals relies on the assumption, stated without proof, that the smallest and largest entries $L$ and $U$ of the shift operator remain fixed with $0<L\le S_{mn}\le U<1$ as the neighborhood size $N_m$ grows; this need not hold for the standard alternating row-column normalizations of growing graphs.

Editorial extensions

If this is right

  • A doubly stochastic graph shift preserves the mean of any graph signal exactly, so repeated shifts act as diffusion toward a uniform signal without changing the baseline level.
  • For i.i.d. random graph signals on graphs with growing incoming neighborhoods, the expected power of the shifted signal converges to $\mu^2$, giving an asymptotic isometry that ordinary adjacency and Laplacian shifts lack.
  • For locally stationary signals with within-neighborhood correlation $\rho$, the expected power after a shift is asymptotically bounded by $\mu^2+\rho\sigma^2 (L+U)^2/(4LU)$, so shifting neither amplifies nor destroys signal energy beyond a controlled factor.
  • Any graph filter of the form $y=\sum_{k=0}^K h_k S^k x$ built on this shift is bounded in $L_1,L_2,L_\infty$ by $\sum_k |h_k|\,\|x\|_p$, making filter design and frequency-response reasoning safer.
  • In the multi-sensor example, using the shift as a spatial expectation operator recovers a temperature field from noisy sensors with a 5.8 dB SNR gain, demonstrating the practical role of the operator as a denoiser.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the isometry proof depends on fixed bounds $L$ and $U$, the result is best read as a statement about graph families whose doubly stochastic normalizations keep every entry bounded away from $0$ and $1$; testing random geometric or power-law graphs would reveal how wide that class actually is.
  • The AM-GM reading of the bias term suggests a graph-design principle the paper only hints at: adding vertices or rewiring so that edge weights within a neighborhood become more homogeneous tightens the bound, and this could be turned into an explicit sensor-placement or edge-weight optimization.
  • Since the variance bound uses only second-order moments and the doubly stochastic structure, the boundedness story should extend to non-Gaussian and heavy-tailed signals, a testable variant the paper does not pursue.
  • Because a doubly stochastic matrix is also the averaging matrix used in consensus algorithms, the result quantifies how much averaging variance remains when each agent's neighborhood grows, which could inform convergence-rate analyses of distributed estimation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript proposes a class of doubly stochastic graph shift operators (GSOs) and claims three properties: (i) lower and upper L2-boundedness for locally stationary random graph signals, (ii) L2-isometry for i.i.d. random graph signals in the asymptotic limit of growing incoming neighbourhoods, and (iii) preservation of the graph signal mean. The theoretical development models the shifted vertex signal as a weighted combination of neighbouring vertex random variables, derives an upper bound on the shifted variance via Cauchy-Schwarz and Kantorovich inequalities, and then takes the limit N_m→∞ to obtain variance vanishing for ρ=0 and a finite bound for ρ>0. A numerical example on temperature sensor data is used to illustrate the denoising effect of the proposed shift.

Significance. If the asymptotic consistency and isometry results were correct, they would provide a useful and simple class of graph shift operators for graph signal processing, with an appealing connection to Markov chains and Sinkhorn-Knopp normalization. The paper also correctly observes that doubly stochastic matrices preserve the mean of any graph signal and preserve the L1 norm of nonnegative signals via the Birkhoff-von Neumann decomposition. However, the central asymptotic claims---variance vanishing for i.i.d. signals and the resulting 'L2-isometry'---are not valid for the stated class of all doubly stochastic matrices. The flaw is load-bearing: it affects Eqs. (19)-(21) and (24), which are the main advertised contributions. Because the error can only be repaired by adding substantive new assumptions and redefining the class of operators, the paper in its current form does not establish its fundamental claims.

major comments (3)
  1. [III-B, Eqs. (15)-(19), Remark 6] The limit in Eq. (19) rests on Remark 6, which asserts that the lower and upper bounds L and U in (17) are invariant to the neighbourhood size N_m. This is not proven and is in fact inconsistent with N_m→∞. For any row m, 1 = ∑_{n∈V_m} S_mn ≥ N_m L, so any sequence of doubly stochastic matrices with N_m→∞ must have L = L(N_m) → 0. Hence the Kantorovich constant K=(L+U)^2/(4LU) depends on N_m, and the replacement of K by a fixed constant in the passage from (18) to (19) is unjustified. Consequently the variance bound (19), the consistency claim (20), and the boundedness results (21) and (24) are not established for the stated class of doubly stochastic GSOs.
  2. [III-B, Remark 7 and Eq. (20)] The claimed asymptotic consistency for i.i.d. signals is false for the stated class. Consider the N×N doubly stochastic matrix S with S_11=1/2, S_1n=S_n1=1/(2(N-1)) for n>1, and S_jn=(1-1/(2(N-1)))/(N-1) for j,n≥2. For i.i.d. vertex signals with variance σ^2, the shifted variance at vertex 1 is σ^2(1/4 + 1/(4(N-1))), which tends to σ^2/4, not 0. This directly contradicts Eq. (20) and Remark 7. Additional hypotheses, such as uniform vanishing of the maximum row entry, are required for the consistency result to hold; they are absent from the paper.
  3. [III-D, Remark 9 and Eq. (24)] The description of Eq. (24) as an 'L2-isometry' is incorrect. The original vertex signal has E{x_m^2}=μ^2+σ^2, whereas the limiting expected power of the shifted signal is μ^2. The operator is therefore norm-contracting in expectation, not norm-preserving; the result describes convergence of S(x_m) to the constant mean μ, a projection, rather than an isometry. This terminology appears in the abstract, introduction, and Remark 9, and materially misrepresents the mathematical content even under additional assumptions that would make the variance vanish.
minor comments (4)
  1. [III-B, Remark 7] The limit subscript 'N_n→∞' should read 'N_m→∞'; the neighbourhood size of vertex m is the quantity being increased.
  2. [II-C, Remark 2] The statement that ‖S‖_2=1 for every doubly stochastic matrix does not follow merely from the largest eigenvalue being equal to 1; it follows from the Birkhoff-von Neumann decomposition used in Remark 4 (or from the convexity of the spectral norm on permutation matrices). The argument as written is incomplete.
  3. [II-C, Remark 4] The L1-isometry statement should explicitly restrict to nonnegative graph signals; for signed signals the L1 norm of Sx can be strictly smaller than that of x, as the mean-preservation argument in Remark 3 shows.
  4. [IV, Numerical example] The example does not report the values of N_m, L, or U for the Sinkhorn-Knopp normalized matrix, so the connection between the theoretical bounds and the demonstrated 5.8 dB SNR gain remains purely illustrative rather than quantitative.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main bounds follow from definitions and standard inequalities; Remark 6 is an unproven assumption but not a circular step.

full rationale

The derivation chain is self-contained in the sense relevant to circularity: the unbiasedness result in Eq. (12) is a direct consequence of the row-stochasticity definition in Eq. (4); the variance bound in Eq. (15) follows from Cauchy-Schwarz; the bound in Eq. (18) follows from the Kantorovich inequality applied to the stated entrywise bounds in Eq. (17); and the lower bound in Eq. (23) is Jensen's inequality. No parameter is fitted to data and then renamed a prediction, and no conclusion is reduced to itself by construction. The self-citations [5] and [6] appear in the introduction and the numerical example but are not load-bearing for the central boundedness or consistency arguments. The paper does rely on Remark 6, which asserts without proof that the bounds L and U are invariant to the neighbourhood size N_m; this is a substantive mathematical-support gap and potentially false for Sinkhorn-Knopp normalized matrices, but an unsupported or even false assumption is not the same as a circular derivation. Similarly, calling Eq. (24) an 'L2-isometry' is a misleading label because the unshifted signal has second moment mu^2+sigma^2, but this is a correctness/terminology issue rather than a self-referential reduction of the result to its input. Overall, no significant circularity is present; the concerns are about validity of assumptions and terminology, not circular reasoning.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claims rest on a locally stationary second-order model, the assumption that GSO entries are bounded away from 0 and 1 with L and U fixed as neighborhoods grow, and the existence of a doubly stochastic normalization. L and U are not estimated; they are assumed so that the Kantorovich inequality yields a vanishing bound. No new physical entities are introduced.

free parameters (2)
  • L
    Lower bound on GSO entries, assumed strictly positive and invariant to neighborhood size (eq. 17, Remark 6). Not fitted to data; introduced as an assumption for the Kantorovich bound.
  • U
    Upper bound on GSO entries, assumed strictly less than 1 and invariant to neighborhood size (eq. 17, Remark 6). Not fitted to data; introduced as an assumption for the Kantorovich bound.
assumptions (4)
  • ad hoc to paper The doubly stochastic shift operator entries satisfy 0 < L ≤ S_mn ≤ U < 1 with L and U independent of N_m.
    Remark 6 states this invariance without proof; it is needed for the limit in eq. (19) to have a constant coefficient. For general doubly stochastic matrices, entries can be 1 or scale with N_m.
  • domain assumption For each neighborhood V_m, vertex signals share identical mean, variance, and pairwise correlation (Section II-B, eqs. 2 and 3).
    This defines the locally stationary model in the vertex domain; the Gaussian assumption is not used in the derivations.
  • standard math The Kantorovich inequality (eq. 16) applies with L and U as the min and max of the weights in each row.
    Standard inequality, correctly cited; it requires the weights to lie in a bounded interval.
  • domain assumption The Sinkhorn-Knopp algorithm converges to a doubly stochastic matrix for the considered weight matrices.
    Algorithm 1 requires convergence, which is not guaranteed for all W; the paper assumes it without discussing conditions.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A Class of Doubly Stochastic Shift Operators for Random Graph Signals and their Boundedness." pith.science (2026). https://pith.science/paper/2BXV7IR2

@misc{pith2026190801596,
  author       = {Pith},
  title        = {Pith review of: A Class of Doubly Stochastic Shift Operators for Random Graph Signals and their Boundedness},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2BXV7IR2}},
  note         = {Machine review of arXiv:1908.01596}
}
abstract

A class of doubly stochastic graph shift operators (GSO) is proposed, which is shown to exhibit: (i) lower and upper $L_{2}$-boundedness for locally stationary random graph signals; (ii) $L_{2}$-isometry for \textit{i.i.d.} random graph signals with the asymptotic increase in the incoming neighbourhood size of vertices; and (iii) preservation of the mean of any graph signal. These properties are obtained through a statistical consistency analysis of the graph shift, and by exploiting the dual role of the doubly stochastic GSO as a Markov (diffusion) matrix and as an unbiased expectation operator. Practical utility of the class of doubly stochastic GSOs is demonstrated in a real-world multi-sensor signal filtering setting.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references · 38 canonical work pages

  1. [1]

    Discrete Signal Processing on Graphs,

    A. Sandryhaila and J. M. F. Moura, “Discrete Signal Processing on Graphs,” IEEE Transactions on Signal Processing , vol. 61, no. 7, pp. 1644–1656, 2013

    work page 2013

  2. [2]

    The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains,

    D. I. Shuman, S. K. Narang, P. Frossard, A. Ortega, and P. Van- dergheynst, “The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains,” IEEE Signal Processing Magazine, vol. 30, pp. 83– 98, 2013

    work page 2013

  3. [3]

    Discrete Signal Processing on Graphs: Sampling Theory,

    S. Chen, R. Varma, A. Sandryhaila, J. Kova ˇcevi´c, J. M. F. Moura, and P. Vandergheynst, “Discrete Signal Processing on Graphs: Sampling Theory,” IEEE Transactions on Signal Processing , vol. 63, no. 24, pp. 6510–6523, 2015

    work page 2015

  4. [4]

    Graph Signal Processing: Overview, Challenges, and Ap- plications,

    A. Ortega, P. Frossard, J. Kova ˇcevi´c, J. M. F. Moura, and P. Van- dergheynst, “Graph Signal Processing: Overview, Challenges, and Ap- plications,” In Proceedings of the IEEE , vol. 106, no. 5, pp. 808–828, 2018

    work page 2018

  5. [5]

    Graph Signal Processing – Part I: Graphs, Graph Spectra, and Spectral Clustering,

    L. Stankovi ´c, D. P. Mandic, M. Dakovi ´c, M. Brajovi ´c, B. Scalzo Dees, and T. Constantinides, “Graph Signal Processing – Part I: Graphs, Graph Spectra, and Spectral Clustering,” arXiv:1907.03467, 2019

    arXiv 1907

  6. [6]

    Understanding the Basis of Graph Signal Processing via an Intuitive Example-Driven Approach,

    L. Stankovi ´c, D. P. Mandic, M. Dakovi ´c, I. Kisil, E. Sejdi ´c, and A. G. Constantinides, “Understanding the Basis of Graph Signal Processing via an Intuitive Example-Driven Approach,” IEEE Signal Processing Magazine, vol. 36, no. 6, pp. 133–145, 2019

    work page 2019

  7. [7]

    Optimal Graph-Filter Design and Applications to Distributed Linear Network Operators,

    S. Segarra, A. G. Marques, and A. Ribeiro, “Optimal Graph-Filter Design and Applications to Distributed Linear Network Operators,” IEEE Transactions on Signal Processing , vol. 65, no. 15, pp. 4117– 4131, 2017

    work page 2017

  8. [8]

    Stationary Graph Processes and Spectral Estimation,

    A. G. Marques, S. Segarra, G. Leus, and A. Ribeiro, “Stationary Graph Processes and Spectral Estimation,” IEEE Transactions on Signal Processing, vol. 65, no. 22, pp. 5911–5926, 2017

    work page 2017

Show all 39 references
  1. [9]

    A Unified View of Diffusion Maps and Signal Processing on Graphs,

    A. Heimowitz and Y . C. Eldar, “A Unified View of Diffusion Maps and Signal Processing on Graphs,” In Proceedings of the International Conference on Sampling Theory and Applications (SampTA) , pp. 308– 312, 2017

    2017

  2. [10]

    On the Shift Operator, Graph Frequency, and Optimal Filtering in Graph Signal Processing,

    A. Gavili and X. P. Zhang, “On the Shift Operator, Graph Frequency, and Optimal Filtering in Graph Signal Processing,” IEEE Transactions on Signal Processing , vol. 65, no. 23, pp. 6303–6318, 2017

    2017

  3. [11]

    Translation on Graphs: An Isometric Shift Operator,

    B. Girault, P. Goncalves, and E. Fleury, “Translation on Graphs: An Isometric Shift Operator,” IEEE Signal Processing Letters , vol. 22, no. 12, pp. 2416–2420, 2015

    2015

  4. [12]

    Translation and Stationarity for Graph Signals,

    ——, “Translation and Stationarity for Graph Signals,” [Research Re- port] RR-8719, ´Ecole Normale Sup ´erieure de Lyon, INRIA , 2015

    2015

  5. [13]

    Stationary Graph Signals using an Isometric Graph Trans- lation,

    B. Girault, “Stationary Graph Signals using an Isometric Graph Trans- lation,” In Proceedings of the European Signal Processing Conference , pp. 1516–1520, 2015

    2015

  6. [14]

    Localization Bounds for the Graph Translation,

    B. Girault, P. Goncalves, S. S. Narayanan, and A. Ortega, “Localization Bounds for the Graph Translation,” In Proceedings of the IEEE Global Conference on Signal and Information Processing , pp. 331–335, 2016

    2016

  7. [15]

    Stationary Signal Processing on Graphs,

    N. Perraudin and P. Vandergheynst, “Stationary Signal Processing on Graphs,” IEEE Transactions on Signal Processing , vol. 65, no. 13, pp. 3462–3477, 2017

    2017

  8. [16]

    Ergodicity in Stationary Graph Processes: A Weak Law of Large Numbers,

    F. Gama and A. Ribeiro, “Ergodicity in Stationary Graph Processes: A Weak Law of Large Numbers,”IEEE Transactions on Signal Processing, vol. 67, no. 10, pp. 2761–2774, 2019

    2019

  9. [17]

    Towards a Definition of Local Stationarity for Graph Signals,

    B. Girault, S. S. Narayanan, and A. Ortega, “Towards a Definition of Local Stationarity for Graph Signals,” In Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 4139–4143, 2017

    2017

  10. [18]

    Local Stationarity of Graph Signals: Insights and Experiments,

    ——, “Local Stationarity of Graph Signals: Insights and Experiments,” In Proceedings of the SPIE Conference on Optical Engineering + Applications, pp. 1–17, 2017

    2017

  11. [19]

    On the Kullback-Leiber Information Divergence of Lo- cally Stationary Processes,

    R. Dahlhaus, “On the Kullback-Leiber Information Divergence of Lo- cally Stationary Processes,” Stochastic Processes and their Applications, vol. 62, pp. 139–168, 1996

    1996

  12. [20]

    R. B. Bapat and T. E. S. Raghavan, Non-negative Matrices and Appli- cations. Cambridge University Press, 1997

    1997

  13. [21]

    Tres Observaciones sobre el Algebra Lineal,

    G. Birkhoff, “Tres Observaciones sobre el Algebra Lineal,” Universidad Nacional de Tucum ´an, Revista. Serie A , vol. 5, pp. 147–151, 1946

    1946

  14. [22]

    B. C. Arnold, Majorization and the Lorenz Order: A Brief Introduction . Springer-Verlag, 1987

    1987

  15. [23]

    Diffusion Maps,

    R. R. Coifman and S. Lafon, “Diffusion Maps,” Applied and Computa- tional Harmonic Analysis , vol. 21, pp. 5–30, 2006

    2006

  16. [24]

    A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices,

    R. Sinkhorn, “A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices,” The Annals of Mathematical Statistics , vol. 35, pp. 876–879, 1964

    1964

  17. [25]

    Concerning nonnegative Matrices and Doubly Stochastic Matrices,

    R. Sinkhorn and P. Knopp, “Concerning nonnegative Matrices and Doubly Stochastic Matrices,” Pacific Journal of Mathematics , vol. 21, pp. 343–348, 1967

    1967

  18. [26]

    The Sinkhorn-Knopp Algorithm: Convergence and Ap- plications,

    P. A. Knight, “The Sinkhorn-Knopp Algorithm: Convergence and Ap- plications,” SIAM Journal on Matrix Analysis and Applications , vol. 30, no. 1, pp. 261–275, 2008

    2008

  19. [27]

    On a Least Squares Adjustment of a Sampled Frequency Table when the Expected Marginal Totals are Known,

    W. E. Deming and F. F. Stephan, “On a Least Squares Adjustment of a Sampled Frequency Table when the Expected Marginal Totals are Known,” The Annals of Mathematical Statistics , vol. 11, no. 4, pp. 427– 444, 1940

    1940

  20. [28]

    An Iterative Method of Adjusting Sample Frequency Tables when Expected Marginal Totals are Known,

    F. F. Stephan, “An Iterative Method of Adjusting Sample Frequency Tables when Expected Marginal Totals are Known,” The Annals of Mathematical Statistics, vol. 13, no. 2, pp. 166–178, 1942

    1942

  21. [29]

    A Unifying Approach to Hard and Proba- bilistic Clustering,

    R. Zass and A. Shashua, “A Unifying Approach to Hard and Proba- bilistic Clustering,” In Proceedings of the International Conference on Computer Vision, pp. 1–8, 2005

    2005

  22. [30]

    Doubly Stochastic Normalization for Spectral Clustering,

    ——, “Doubly Stochastic Normalization for Spectral Clustering,” In Proceedings of the Conference on Neural Information Processing Sys- tems (NIPS), pp. 1569–1576, 2006

    2006

  23. [31]

    Structured Doubly Stochastic Matrix for Graph Based Clustering: Structured Doubly Stochastic Matrix,

    F. Wang, P. Li, and A. C. Konig, “Structured Doubly Stochastic Matrix for Graph Based Clustering: Structured Doubly Stochastic Matrix,” In Proceedings of the Conference on Knowledge Discovery and Data Mining, pp. 1245–1254, 2016

    2016

  24. [32]

    Robust Multi-Class Transductive Learning with Graphs,

    W. Liu and W. Chang, “Robust Multi-Class Transductive Learning with Graphs,” In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 381–388, 2009

    2009

  25. [33]

    Learning a Bi-Stochastic Data Similarity Matrix,

    F. Wang, P. Li, and A. C. Konig, “Learning a Bi-Stochastic Data Similarity Matrix,” In Proceedings of the IEEE International Conference on Data Mining , pp. 551–560, 2010

    2010

  26. [34]

    Unsupervised and Semi- Supervised Learning via 𝓁1-Norm Graph,

    F. Nie, H. Wang, H. Huang, and C. H. Ding, “Unsupervised and Semi- Supervised Learning via 𝓁1-Norm Graph,” In Proceedings of the IEEE International Conference on Computer Vision , pp. 2268–2273, 2011

    2011

  27. [35]

    Forging The Graphs: A Low Rank and Positive Semidefinite Graph Learning Approach,

    D. Luo, H. Huang, F. Nie, and C. H. Ding, “Forging The Graphs: A Low Rank and Positive Semidefinite Graph Learning Approach,” In Proceedings of the Conference on Neural Information Processing Systems (NIPS), pp. 2969–2977, 2012

    2012

  28. [36]

    Consensus + Innovations Distributed Infer- ence over Networks: Cooperation and sensing in Networked Systems,

    S. Kar and J. M. F. Moura, “Consensus + Innovations Distributed Infer- ence over Networks: Cooperation and sensing in Networked Systems,” IEEE Signal Processing Magazine , vol. 30, no. 3, pp. 99–109, 2013

    2013

  29. [37]

    Graph Signal Processing: Filter Design and Spectral Statistics,

    S. Kruzick and J. M. F. Moura, “Graph Signal Processing: Filter Design and Spectral Statistics,” In Proceedings of the IEEE International Work- shop on Computational Advances in Multi-Sensor Adaptive Processing , pp. 1–5, 2017

    2017

  30. [38]

    Some Applications of Doubly Stochastic Matrices,

    R. A. Brualdi, “Some Applications of Doubly Stochastic Matrices,” Linear Algebra and its Applications , vol. 107, pp. 77–100, 1988

    1988

  31. [39]

    Functional Analysis and Applied Mathematics,

    L. V . Kantorovich, “Functional Analysis and Applied Mathematics,” Uspekhi Matematicheskikh Nauk , vol. 3, no. 6, pp. 89–185, 1948

    1948

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.