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REVIEW 3 major objections 4 minor 44 references

Optimal Bregman quantization : existence and uniqueness of optimal quantizers revisited

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Bregman quantization admits a unique optimal codebook for strongly unimodal distributions when the generator's second derivative is log-concave or log-convex.

desk verdict The existence theorem is a real improvement, but the uniqueness theorem collapses for log-convex F'' on the false bound ψ≤1. read the letter →

arxiv 2506.01746 v1 pith:3X2PDDO6 submitted 2025-06-02 math.PR

Guillaume Boutoille , Gilles Pagès This is my paper
classification math.PR MSC 60E0562H3094A34
keywords BregmandivergenceoptimalquantizationTrushkinuniquenesstheoremstronglyunimodaldistributionlog-concavedensitystationaryquantizermasterequationk-meansclustering
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

This paper revisits optimal quantization when the distortion is measured by a Bregman divergence $\phi_F$ rather than a power of a norm, and it tries to establish two things: an existence theorem under lighter hypotheses on the strictly convex generator $F$, and a one-dimensional uniqueness theorem. The uniqueness result, modelled on Trushkin's theorem, states that if $P$ is strongly unimodal with a log-concave density on $[a,b]$, if $F$ and $F''$ are positive, and if $F''$ is either log-concave or log-convex, then for every level $n \geq 1$ the master equation has at most one solution in the ordered simplex; combined with existence, there is exactly one optimal Bregman quantizer. If true, the practical payoff is that Bregman k-means type algorithms face no spurious local minima on such problems: different initial codebooks should converge to the same grid. The paper also records structural properties such as $P$-negligible Voronoi boundaries, stationarity, and symmetry of optimal quantizers for symmetric $P$ with even $F''$.

What carries the argument

The load-bearing object is the one-dimensional Bregman bisector $\varphi(u,v) = u + \phi_F(u,v)/(F'(v)-F'(u)) = v - \phi_F(v,u)/(F'(v)-F'(u))$, which gives the cell boundaries of a codebook, together with the ratio $\psi(u,v) = \partial_u\varphi(u,v) + \partial_v\varphi(u,v) = [F''(u)\phi_F(u,v)+F''(v)\phi_F(v,u)]/(F'(v)-F'(u))^2$. The uniqueness proof is organized around the inequality $\psi(u,v) \leq 1$, derived from log-concavity or log-convexity of $F''$. This inequality performs two tasks: it shows, via a Gershgorin lemma on the tridiagonal Hessian of the distortion, that every stationary quantizer is a strict local minimum; and it shows that a sufficiently small gradient-descent step $I - \varepsilon\nabla G_n$ keeps codebooks inside the ordered simplex $S_n(a,b)$. The Mountain Pass Lemma then forces at most one stationary point, and existence from Theorem 5.1 upgrades that to uniqueness of the optimum.

What would settle it

Evaluate $\psi(u,v)$ numerically for the generators listed in Section 8.1; in particular, for $F(x)=-\log x$ (Itakura–Saito, a log-convex example) at $u=1$, $v=2$, the quotient exceeds 1, contradicting the key bound used in the proof. A direct test of the theorem itself is to run the Bregman-Lloyd iteration from many initial codebooks for a strongly unimodal $P$ with such a generator and check whether two distinct stationary $n$-tuples appear for some $n$.

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Extended reading notes

Core claim

The central claim is the Bregman analogue of Trushkin's uniqueness theorem. Assume $P$ has a log-concave density on its support $[a,b]$ and the generator $F$ satisfies $F>0$ and $F''>0$ on $(a,b)$, with $\log F''$ either concave or convex. Then, for every level $n \geq 1$, there is at most one $n$-tuple in $S_n(a,b)$ solving the Bregman master equation $P(C_i(x))x_i = \int_{C_i(x)} \xi\, dP(\xi)$, $i=1,\ldots,n$; because the authors' existence theorem guarantees at least one such tuple, it is the unique Bregman-optimal quantizer. The same statement covers both the curvature-log-concave case (squared norm, power losses, softplus, exponential, I-divergence) and the curvature-log-convex case (Itakura–Saito, logistic), and the proof does not rely on symmetry of $\phi_F$.

Load-bearing premise

The load-bearing premise is that the boundary-ratio function $\psi(u,v)$ never exceeds 1 when $F''$ is log-concave or log-convex, for this bound is what makes every stationary quantizer a strict local minimum and keeps gradient steps inside the ordered simplex.

Editorial extensions

If this is right

  • For a strongly unimodal distribution and an admissible $F$, the Bregman-Lloyd fixed-point iteration has a unique fixed codebook at every level $n$; no spurious stationary quantizers exist to trap local search.
  • Under the same assumptions the optimal quantizer is the unique minimizer of the distortion $G_n$, and the optimal value is $E[F(X)] - E[F(\hat X^{x(n)})]$.
  • For symmetric $P$ with even $F''$, the unique optimal grid is symmetric: $x_i = -x_{n+1-i}$ for all $i$, so the search for optimal codebooks can be restricted to the symmetric subspace.
  • The existence theorem applies under a milder boundary condition at infinity (in the quadratic case), covering generators such as softplus, exponential, and logistic losses that earlier existence results did not cover.
  • The distortion sequence $G_n(x^{(n)})$ decreases and tends to $0$ as $n \to \infty$, so larger codebooks monotonically improve the Bregman approximation.

Reading between the lines

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

  • The proof's key object, the boundary-ratio function $\psi$, is computable for any smooth generator; a grid search over $(u,v)$ would give practitioners a quick certification test for whether a new divergence satisfies the theorem's structural condition.
  • One extension the paper leaves open is to carry the same Gershgorin/Mountain-Pass argument into the $r>2$ Bregman-quantization setting of Section 7; the master equation there is a tilted conditional expectation, and a $\psi_r$ bound depending on $r$ would give an explicit uniqueness criterion for Bregman medians.
  • Because the optimal codebook is unique, the identity $e_{2,n}(P,\phi_F)^2 = E[F(X)] - E[F(\hat X^{x(n)})]$ becomes a principled model-selection score for comparing Bregman divergences on the same data, an application the paper does not develop.
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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 paper revisits L^r-optimal quantization (r ≥ 2) with respect to a Bregman divergence generated by a strictly convex function F. The first main contribution is an existence theorem for optimal quantizers under a condition at infinity, Eq. (24), which is weaker than the boundary conditions used in prior work. The second main contribution is a one-dimensional uniqueness theorem 'à la Trushkin' (Theorem 8.1): for strongly unimodal distributions and twice-differentiable F with positive F'' that is either log-concave or log-convex, there is at most one solution to the master equation (25), hence exactly one Bregman optimal quantizer. The paper also proves differentiability of the distortion, stationarity of optimal quantizers, invariance of the simplex under a gradient step, and includes several examples and numerical illustrations.

Significance. If correct, the existence result is a genuine improvement: it replaces pointwise conditions on the boundary of U by a single liminf/sup condition at infinity in the unbounded case, and the proof via Alexandroff compactification and induction is a rigorous and valuable extension of earlier work. The uniqueness theorem would likewise be a significant extension of Trushkin's classical result to Bregman divergences, covering such practically used losses as Itakura–Saito, I-divergence, logistic, and softplus divergences. However, the paper's central uniqueness theorem is stated for both log-concave and log-convex F''; the proof of the log-convex branch rests on a false inequality, so the theorem as stated is not established. The log-concave branch and the existence theorem may well be correct, but the main advertised claim of the paper is broader than what the proof supports.

major comments (3)
  1. [Section 8.3, Step 3, Eq. (53)–(54)] The inequality ψ(u,v) ≤ 1 is asserted for both log-concave and log-convex F''. For the log-convex case the sign of the cross derivative is handled incorrectly. From ∂²_ur,vr Φ(u,v) = F''(u)F''(v)[(log F'')'_r(u) − (log F'')'_r(v)](v−u) ≤ 0 (since (log F'')' is non-decreasing when log F'' is convex), one obtains that u ↦ ∂_vr Φ(u,v) is non-increasing in u. Hence for u ≤ v, ∂_vr Φ(u,v) ≥ ∂_vr Φ(v,v) = 0, not ≤ 0. The paper instead concludes ∂_vr Φ(u,v) ≤ ∂_vr Φ(u,u) = 0 and therefore Φ(u,v) ≤ Φ(u,u) = 0, which is the opposite of the correct implication. The claimed ψ ≤ 1 is in fact false for a listed log-convex example: for F(x) = −log x (Itakura–Saito), direct computation at u = 1, v = 2 gives ψ(1,2) = [F''(1)ϕ(1,2) + F''(2)ϕ(2,1)]/(F'(2)−F'(1))² ≈ (0.1931+0.0767)/0.25 ≈ 1.079 > 1, i.e., Φ(1,2) > 0. Thus Eq. (54) is false in the log-convex branch.
  2. [Section 8.3, Step 3 and Theorem 8.1] The false bound ψ ≤ 1 is load-bearing. It is used twice in the proof of Theorem 8.1 for the log-convex case: first, through the Gershgorin row sums (48)–(51), to conclude that every stationary quantizer is a strict local minimum; second, in the argument after Eq. (43), to show that Id − ε∇G_n maps S_n(a,b) into itself. Without ψ ≤ 1, neither conclusion follows for log-convex F'', and the Mountain Pass argument does not establish uniqueness. Since Theorem 8.1 is stated for 'either log-concave or log-convex' F'', the theorem as stated is unsupported. The log-concave branch may be salvageable, but the log-convex branch—which is explicitly advertised in the abstract and illustrated by the Itakura–Saito and logistic examples—has no valid proof in the current manuscript.
  3. [Abstract and Introduction (also Section 8.1)] Because the log-convex branch of Theorem 8.1 collapses, the paper's claimed scope is too broad. The abstract states 'divergences generated by strictly convex functions whose third derivative is either strictly log-convex or log-concave', and Section 8.1 lists Itakura–Saito and logistic divergences as log-convex examples. The authors need either to provide a correct proof covering these cases or to restrict the uniqueness theorem to log-concave F'' (and adjust the abstract, Theorem 8.1, and the examples accordingly). As it stands, the central result of the paper is not proven in a substantial part of its stated domain.
minor comments (4)
  1. [Abstract and throughout] The abstract contains several typos: 'exsistence', 'quantizaers', 'whiose thire dervative', and 'gerated'. These should be corrected.
  2. [Section 8.3, Step 3, log-convex case] In the log-convex case the text says 'v ↦ Φ(u,v) is non-decreasing on [v,b)'; the preceding sentence implies the opposite, namely non-increasing. This is more than a wording issue because the correct monotonicity reverses the inequality anyway, but the wording should be fixed in any revision.
  3. [Section 3.1, Proposition 3.2 proof] 'Cauchy-Schwartz' should be 'Cauchy-Schwarz'.
  4. [Section 5.2, Examples] In the logistic similarity measure, the formula uses '1 − v' in the last term; this should be '1 − x' to match the definition of the logistic divergence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Bregman quantization existence and uniqueness results are derived from stated assumptions and standard external lemmas; self-citations are contextual, not load-bearing tautologies.

full rationale

The paper's central claims—existence of Bregman optimal quantizers under lighter assumptions (Theorems 5.1 and 7.1) and one-dimensional uniqueness for strongly unimodal P and log-concave or log-convex F'' (Theorem 8.1)—are proved from the definitions of the Bregman divergence and distortion (Eqs. (3), (9)), the moment assumption (8), and the master equation (25). No parameter is fitted, no normalization is chosen to force the conclusion, and no target result is assumed among the hypotheses. The proof tools imported from earlier work, notably the compact Mountain Pass Lemma (Theorem 8.2(b), cited to [14], [5], [16]) and standard quantization background ([18], [16], [9]), are general external theorems whose assumptions do not include the uniqueness statement; citing them is ordinary use of the literature, not circularity. The companion thesis [2] is mentioned for numerical implementation and further examples, not as the proof of Theorem 8.1. The reader's objection that the bound psi(u,v) <= 1 fails for log-convex F'' (e.g. F(x) = -log x) concerns the validity of Step 3 of the proof, not whether the conclusion was assumed as an input; hence it is a correctness risk, not a circularity. Accordingly there is no circular step to report.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted and no new entities are introduced. The central results are pure existence and uniqueness theorems, so the ledger consists only of the analytic and domain assumptions stated in the theorems.

assumptions (5)
  • domain assumption P is strongly unimodal (log-concave density) on [a,b]
    Used throughout Section 8 to get log-concavity properties of h and to justify integration by parts with h'_r/h non-increasing.
  • domain assumption F'' > 0 on (a,b) and log F'' either convex or concave
    This is the central hypothesis of Theorem 8.1 and drives the sign of the cross-partial derivative of Phi in Section 8.3 Step 3.
  • ad hoc to paper Condition (24): for unbounded U, liminf_{|x| -> inf} phi_F(xi,x) = sup_{x in U} phi_F(xi,x)
    Technical condition introduced in Theorem 5.1 to prevent minimizers escaping to the Alexandroff boundary; the remark notes it is lighter than Fischer's (33).
  • domain assumption Moment and integrability assumption (8): E(|X| or |F(X)|)^{r/2} < infinity
    Assumed for all existence theorems and for the distortion function to be finite.
  • standard math Mountain Pass Lemma and Gershgorin Lemma
    Used in Section 8.2 to conclude that all critical points being strict local minima implies uniqueness. Gershgorin is proved in the paper; the Mountain Pass Lemma is cited from [5,13,14,16,22].

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Pith. "Pith review of Optimal Bregman quantization : existence and uniqueness of optimal quantizers revisited." pith.science (2026). https://pith.science/paper/3X2PDDO6

@misc{pith2026250601746,
  author       = {Pith},
  title        = {Pith review of: Optimal Bregman quantization : existence and uniqueness of optimal quantizers revisited},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3X2PDDO6}},
  note         = {Machine review of arXiv:2506.01746}
}
abstract

In this paper we revisit the exsistence theorem for $L^r$-optimal quantization, $r\ge 2$, with respect to a Bregman divergence: we establish the existence of optimal quantizaers under lighter assumptions onthe strictly convex function which generates the divergence, espcially in the quadratic case ($r=2$). We then prove a uniqueness theorem ``\`a la Trushkin'' in one dimension for strongly unimodal distributions and divergences gerated by strictly convex functions whiose thire dervative is either stictly $\log$-convex or $\log$-concave.

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Reference graph

Works this paper leans on

44 extracted references · 43 canonical work pages

  1. [1]

    make the job

    that ∥ · ∥is differentiable on Rd \ {0}, like Euclidean norms or ℓp-norms, 1 < p <+∞. 5 Remark. In view of what follows with Bregman divergence it is interesting to inspect a variant of the above result as concerns the state space : if the norm under consideration is an Euclidean norm denoted | · |e and P is supported by a (nonempty) closed convex set, sa...

    work page

  2. [2]

    F (x) = x2, U = R, ϕF (ξ, x) = (ξ − x)2

    Regular quadratic similarity measure. F (x) = x2, U = R, ϕF (ξ, x) = (ξ − x)2

    work page

  3. [3]

    F (x) = xa, a >1, U = (0, +∞), and ϕF (ξ, x) = ξa + (a − 1)xa − a ξ xa−1

    Norm–like similarity measure . F (x) = xa, a >1, U = (0, +∞), and ϕF (ξ, x) = ξa + (a − 1)xa − a ξ xa−1

    work page

  4. [4]

    F (x) = − log(x), U = (0, +∞), and ϕF (ξ, x) = log x ξ + ξ x − 1

    Itakura–Saito divergence. F (x) = − log(x), U = (0, +∞), and ϕF (ξ, x) = log x ξ + ξ x − 1

    work page

  5. [5]

    Kullback–Leibler divergence/similarity measure

    I-divergence a.k.a. Kullback–Leibler divergence/similarity measure . F (x) = x log(x), U = (0, +∞) and ϕF (ξ, x) = ξ log ξ x − 1 + x ξ

    work page

  6. [6]

    F (x) = x log x + (1 − x) log(1 − x), U = (0, 1), and ϕF (ξ, x) = ξ log ξ x + (1 − ξ) log 1 − ξ 1 − v

    Logistic similarity measure. F (x) = x log x + (1 − x) log(1 − x), U = (0, 1), and ϕF (ξ, x) = ξ log ξ x + (1 − ξ) log 1 − ξ 1 − v . 21

    work page

  7. [7]

    F (x) = Fa(x) = log(1 + eax)/a, a >0, U = R, and ϕF (ξ, x) = 1 a log 1 + eaξ 1 + eax ! − eax eax + 1(ξ − x)

    Softplus similarity measure (smooth approximation of the RELU function ( 2)). F (x) = Fa(x) = log(1 + eax)/a, a >0, U = R, and ϕF (ξ, x) = 1 a log 1 + eaξ 1 + eax ! − eax eax + 1(ξ − x)

    work page

  8. [8]

    F (x) = Fa(x) = log(cosh(ax)) a , a >0, U = R

    Soft butterfly similarity measure (smooth approximation of the absolute value function) . F (x) = Fa(x) = log(cosh(ax)) a , a >0, U = R. Then, with obvious notations ϕF (ξ, x) = 1 a log cosh(aξ) cosh(ax) ! − tanh (ax)(ξ − x) = 2 ϕSoftPlus2a(ξ, x). The second equality is in fact a consequence of the elementary identity (see Proposition 3.1) log cosh(ax) a ...

    work page

Show all 44 references
  1. [9]

    Fρ(x) = eax, a ∈ R, U = R, ϕFρ(ξ, x) = ϕF1(a ξ, a x) where ϕF1(ξ, x) = eξ − ex − ev(ξ − x)

    Exponential loss . Fρ(x) = eax, a ∈ R, U = R, ϕFρ(ξ, x) = ϕF1(a ξ, a x) where ϕF1(ξ, x) = eξ − ex − ev(ξ − x). However, note that 6. (and 7.) do not fulfill the assumption contained e.g. in [7] to guarantee existence of optimal quantizers for such Bregman divergences, that is ...

  2. [10]

    F (x) = |x|2, x ∈ U = Rd

    Regular quadratic similarity measure. F (x) = |x|2, x ∈ U = Rd

  3. [11]

    F (x) = x∗Sx, x ∈ U = Rd, S ∈ S++(d, R) := S +(d, R) ∩ GL(d, R) (symmetric and positive definite), and ϕF (ξ, x) = (ξ − x)∗S(ξ − x) := |ξ − x|2 S

    (Squared) Mahalanobis distance . F (x) = x∗Sx, x ∈ U = Rd, S ∈ S++(d, R) := S +(d, R) ∩ GL(d, R) (symmetric and positive definite), and ϕF (ξ, x) = (ξ − x)∗S(ξ − x) := |ξ − x|2 S

  4. [12]

    f -marginal divergence (additive similarity measure) . F (x1, . . . , xd) = Pd i=1 f (xi), f strictly convex on V , is defined on U = V d, and for every x = (x1, . . . , xd), ξ = (ξ1, . . . , ξd) ∈ U , ϕF (ξ, x) = dX i=1 ϕf (ξi, xi)

  5. [13]

    f -marginal divergence (multiplicative similarity measure) . F (x1, . . . , xd) = Qd i=1 f (xi), f strictly convex on V , is defined on U = V d, and for every x = (x1, . . . , xd), ξ = (ξ1, . . . , ξd) ∈ U , ϕF (ξ, x) = dX i=1 ϕf (ξi, xi) Y j̸=i ϕf (ξi, xj)

  6. [14]

    For REctified Linear Unit. 22

  7. [15]

    Soft max marginal f -divergence as similarity measure . F (x1, . . . , xd) = Fλ(x1, . . . , xd) = 1 λ log Pd i=1 eλf (xi) , f strictly convex on V , is defined on U = V d and for every x = (x1, . . . , xd), ξ = (ξ1, . . . , ξd) ∈ U d, ϕF (ξ, x) = Fλ(ξ) − Fλ(x) − P 1≤i≤d f ′(xi...

  8. [16]

    ∂GF r,n ∂xi (x) # i=1,...,n = r 2 Diag∇2F (xi) i=1,...,n

    Soft norm similarity measure . F (x1, . . . , xd) = Pd i=1 fλ(ξi, xi), fa(u) = log cosh(λu) /λ, a >0 (soft butterfly similarity measure), is defined onU = Rd and for every x = (x1, . . . , xd), ξ = (ξ1, . . . , ξd) ∈ U , ϕF (ξ, x) = dX i=1 ϕfλ(ξi, xi). Figure 1 – Non-convexity...

    1936

  9. [17]

    F (x) = |x|2 (regular optimal quadratic quantization)

  10. [18]

    F (x) = P i=1,2 log(1 + eaxi)/a (SoftPlus, additive marginals, a = 1)

  11. [19]

    F (x) = P i=1,2 eaxi (exponential, additive marginals) a = 1 and −1

  12. [20]

    F (x) = P i=1,2 xi log xi (Kullback–Leibler)

  13. [21]

    F (x) = − P i=1,2 log xi (Itakura–Sa ¨ ıto) R´ ef´ erences

  14. [22]

    Banerjee, S

    A. Banerjee, S. Merugu, I. S. Dhillon, and J. Ghosh. Clustering with Bregman divergences. J. Mach. Learn. Res., 6 :1705–1749, 2005

    2005

  15. [23]

    Boutoille

    G. Boutoille. Quantification optimale par divergence de Bregman et applications industrielles . PhD thesis, 2024. Adv. Gilles Pag` es, Mathematics, Sorbonne Universit´ e (France), 2024

    2024

  16. [24]

    Bouton and G

    C. Bouton and G. Pag` es. Self-organization and a.s. convergence of the one-dimensional Koho- nen algorithm with non-uniformly distributed stimuli. Stochastic Process. Appl., 47(2) :249– 274, 1993

    1993

  17. [25]

    S. D. Chatterji. Les martingales et leurs applications analytiques. In ´Ecole d’ ´Et´ e de Proba- bilit´ es : Processus Stochastiques (Saint Flour, 1971), pages 27–164. Lecture Notes in Math., Vol. 307. Springer, Berlin, 1973

    1971

  18. [26]

    P. Cohort. Sur quelques problemes de quantification . PhD thesis, Paris 6, 2000

    2000

  19. [27]

    J. A. Cuesta and C. Matr´ an. The strong law of large numbers for k-means and best possible nets of Banach valued random variables. Probability theory and related fields , 78(4) :523–534, 1988

    1988

  20. [28]

    A. Fischer. Quantization and clustering with Bregman divergences. Journal of Multivariate Analysis, 101 :2207–2221, 2010

    2010

  21. [29]

    Gersho and R.M

    A. Gersho and R.M. Gray. Vector Quantization and Signal Compression . The Springer Inter- national Series in Engineering and Computer Science. Springer US, 2012

    2012

  22. [30]

    Graf and H

    S. Graf and H. Luschgy. Foundations of Quantization for Probability Distributions , volume 1730 of Lecture Notes in Mathematics . Springer, Berlin, 2000

    2000

  23. [31]

    Has, A Fischer, and M

    S. Has, A Fischer, and M. Mougeot. Kfc : A clusterwise supervised learning procedure based on the aggregation of distances. Journal of Statistical Computation and Simulation , 91(11) :2307– 2327, 2021

    2021

  24. [32]

    A. K. Jain and R. C. Dubes. Algorithms for clustering data . Prentice Hall Advanced Reference Series. Prentice Hall, Inc., Englewood Cliffs, NJ, 1988

    1988

  25. [33]

    Jourdain and G

    B. Jourdain and G. Pag` es. Optimal dual quantizers of 1D log-concave distributions : uniqueness and Lloyd like algorithm. Journal of Approximation Theory , 267 :105581, 2021

    2021

  26. [34]

    O. Kavian. Introduction ` a la th´ eorie des points critiques et applications aux probl` emes ellip- tiques, volume 13 of Math. Appl. (Berl.) . Paris : Springer-Verlag, 1993. 42

    1993

  27. [35]

    Lamberton and G

    D. Lamberton and G. Pag` es. On the critical points of the 1-dimensional competitive learning vector quantization algorithm. In 4th European Symposium on Artificial Neural Networks, ESANN 1996, Bruges, Belgium, April 24-26, 1996, Proceedings , 1996

    1996

  28. [36]

    M. Liu, C. Belkin. Clustering with bregman divergences : an asymptotic analysis. In D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 29. Curran Associates, Inc., 2016

    2016

  29. [37]

    Luschgy and G

    H. Luschgy and G. Pag` es. Marginal and Functional Quantization of Stochastic Processes . Probability Theory and Stochastic Modelling. Springer, Cham, 2023

    2023

  30. [38]

    MacQueen

    J. MacQueen. Some methods for classification and analysis of multivariate observations. the 5th Berkley Symposium on Mathematical Statistics and Probability , 1967

    1967

  31. [39]

    G. Pag` es. A space quantization method for numerical integration. Journal of Computational and Applied Mathematics , 89(1) :1–38, 1998

    1998

  32. [40]

    G. Pag` es. Introduction to vector quantization and its applications for numerics. ESAIM : Proceedings and Surveys, 48 :29–79, 2015

    2015

  33. [41]

    Pag` es and J

    G. Pag` es and J. Yu. Pointwise convergence of the Lloyd I algorithm in higher dimension. SIAM Journal on Control and Optimization , 54(5) :2354–2382, 2016

    2016

  34. [42]

    Sabin and R

    M. Sabin and R. Gray. Global convergence and empirical consistency of the generalized lloyd algorithm. IEEE Transactions on information theory , 32(2) :148–155, 1986

    1986

  35. [43]

    M. Struwe. Variational Methods, volume 34 of Ergebnisse der Mathematik und ihrer Grenz- gebiete (3) [Results in Mathematics and Related Areas (3)] . Springer-Verlag, Berlin, second edition, 1996. Applications to nonlinear partial differential equations and Hamiltonian systems

    1996

  36. [44]

    Trushkin

    A. Trushkin. Sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions. IEEE Transactions on Information Theory , 28(2) :187–198, 1982. 43 Figure 5 – P = N 1 2 1 ; 1 4 I2 . Bregman functions (from left to right and top ...

    1982

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