On Convolutions, Intrinsic Dimension, and Diffusion Models
Authors: Kin Kwan Leung
Rasa Hosseinzadeh
Gabriel Loaiza-Ganem
Venue: Accepted by TMLR ()

Abstract: The manifold hypothesis asserts that data of interest in high-dimensional ambient spaces, such as image data, lies on unknown low-dimensional submanifolds. Diffusion models (DMs) -- which operate by convolving data with progressively larger amounts of Gaussian noise and then learning to revert this process -- have risen to prominence as the most performant generative models, and are known to be able to learn distributions with low-dimensional support. For a given datum in one of these submanifolds, we should thus intuitively expect DMs to have implicitly learned its corresponding local intrinsic dimension (LID), i.e. the dimension of the submanifold it belongs to. Kamkari et al. (2024b) recently showed that this is indeed the case by linking this LID to the rate of change of the log marginal densities of the DM with respect to the amount of added noise, resulting in an LID estimator known as FLIPD. LID estimators such as FLIPD have a plethora of uses, among others they quantify the complexity of a given datum, and can be used to detect outliers, adversarial examples and AI-generated text. FLIPD achieves state-of-the-art performance at LID estimation, yet its theoretical underpinnings are incomplete since Kamkari et al. (2024b) only proved its correctness under the highly unrealistic assumption of affine submanifolds. In this work we bridge this gap by formally proving the correctness of FLIPD under realistic assumptions. Additionally, we show that an analogous result holds when Gaussian convolutions are replaced with uniform ones, and discuss the relevance of this result.

Published in Transactions on Machine Learning Research (10/2025)
On Convolutions, Intrinsic Dimension, and Diffusion Models
Kin Kwan Leung Layer 6 AI
kk@llayer.cc oi
Rasa Hosseinzadeh
rasa@layer6.ct
Lager 6 Al
Gabriel Loaiza-Ganem
gabricl@layer6..t
Layer 6 Al
Reviewed on OpenReview: https: // openreview. net/ forum? id= #SzBfIte4s
Abstract
The manifold hypothesis asserts that data of interest in high-dimensional ambient spaces, such as image data, lies on unknown low-dimensional submanifolds. Diffusion models (DMs) which operate by convolving data with progressively larger amounts of Gaussian noise and then learning to revert this process have risen prominence as the most performant gener- ative models, and are known to be able to learn distributions with low-dimensional support. For given datum in one of these submanifolds, have implicitly learned its corresponding local intrinsic dimension (LID), i.e. the dimension of the submanifold it belongs to. Kamkari et al. (0024b) recently showed that this is in- deed the case by linking this LID to the rate of change the log marginal densities of the DM with respect the amount added noise, resulting in an LID estimator known as FLIPD. LID estimasors such as FLIPD have a plethora of uses, among others they quantify the complexity of a given datum, and can be used detect outliers, adversarial examples and Al-generated text. FLIPD achieves state-of-tho-art performance at LID estimation, yet its theoretical underpinnings are incomplete since Kankari el al. (2024b) only proved ils correctness under the highly unrealistic assumption of affine submanifolds. In this work we bridge this gap by formally proving the correctness of FLIPD under realistic assump- tions. Additionally, we show that an analogous result holds when Gaussian convolutions are replaced with uniform ones, and discuss the relevance of this result.
should thus intuitively expect DMs to
1 Introduction
The manifold hypothesis (Bengio et. al. 2013) states that high-dimensional data in RD commonly lies on low-dimensional submanifold, or disjoint union thereof (Brown et al. 2023 embedded on RD; this hypothesis has been empirically found to hold for image data (Pope et al. 2021 and other modalities (Cresswell et al. 2022). Given a dataset obeying this hypothesis along with a query datum I, one might want leverage the data to estimate the local intrinsic dimension (LID) of the datum i.c. the dimension of the which we denote as LID(x). At first glance the problem of LID estimation might seem like mere statistical or mathematical curiosity, yet LID estimators have found numerous important applications within machine learning: they effectively quantify image complexity (Kamkari et al. 2024b
submanifold it belongs
are helpful in the detection of outlicrs (Houle et al. 2018 Anderberg ct al. 2024 Kamkari et al.
2024a
Al-generated lext (Tulchinskii et. al.) 2023), memorization [Ross el al. 2025) and adversarial examples (Ma
et al. 2018 and LID estimates of neural network representations are predictive of generalization (Ansuini
et al. 2019 Birdal et al. 2021 Brown al.] 2022 Magai Ayzenberg 2022). However, traditional
LID
estimators rely on pairwise distances or similar quantities (Fukunaga Olsen
1971 Pettis et al.
1979
Grassberger & Procaccia 1983 Levina. & Bickel 2004
MacKay & Ghahramani
2005 Johnsson et al.
2014

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Facco et al., 2017: Bac et al. 2021), J, and they thus typically suffer both from poor scaling datasct size and from the curse of dimensionality.
Generative models aim to learn the distribution p that gave rise to observed data, and when the manifold hypothesis holds, p has low-dimensional support. Research studying generative models through the lens of
the manifold hypothesis has recently proliferated (Dai & Wipf 2019: Zhang et al. 2020; Brehmer & Cranmer
2020
Kim et al.] 2020 Arbel et al. 2021 Caterini al. 2021
Horvat & Pfister 2021; Ross & Cresswell
2021
Cunningham ct al. 2022. Loaiza-Ganenn et al.] 2022a Ross et al.) 2023 Sorrenson et al.] 2024
Loaiza-Gardd Ganem et al. 2024 Ventura et al. 2025), and an important finding from this line of research is that
diffusion models (DMs: Sohl-Dickstein et al.) 2015 Ho et al., 2020, Song et al. 2021) which are state-of-
the-art generative models can successfully learn distributions with low-dimensional support (Pidstrigach)
2022 De Bortoli 2022). When a generative model, such as a DM, correctly learns it must therefore learn the corresponding intrinsic dimensions of its support as well; indeod, various works have shown that LID estimators can be extracted from generative models and that these new estimators avoid the aforementioned
pitfalls of traditional ones (Tempezyk et. al. 2022: Zheng et al. 2022 Horvat & Pfister 2024 Stanezuk et al.
2024
Kamkari et al. 2024b).
FLIPD is a state-of-the-art and highly scalable LID estimator based on DMs that was recently proposed by
Kamkari et al. (2024b). FLIPD is grounded in a result showing that
a log ex(2,)).
LID(z)= D lim 0o 06
where (6) denotes the convolution between p and Gaussian noise with log standard deviation DMs operate by adding Gaussian noise to data and estimating the correspouding score function; the right-hand side of Equation T can be written terms of this score function, which then enables FLIPD to be computed by plugging-in the score function learned by the DM. However, Kamkari et al. (2024b) only proved Equation Eroved[Equatinn in the case where p is supported on an affine submanifold of R this is highly restrictive and unrealistic assumption for typical data of interest such as natural images.
In summary, LID estimation is a relevant problem in machine learning and the current best-performing estimator of LID lacks strong formal justification. In this work we remedy this situation by proving that Equation 1 actually holds when is supported on general disjoint unions of submanifolds of IR,,, thus fully justifying FLIPD. Additionally, we also prove that an analogous result holds when the convolution is carriod out against uniform distribution on ball of log radius 8 (rather than Gaussian with log standard deviation 8). Although this second result is less practically relevant since DMs do not use uniform noise, it is still of interest since it directly links LID(x) to the probability assigned by p to ball around æ, which is quantity that naturally appears implicitly and explicitly throughout machine learning (see e.g. Silverman 1986. Naeem et al. 2020
Bhattacharjee et al. 2023: Kamkari et al. 2024a).
2 Setup and Background
2.1 Setup and Notation
Throughout our work, we will follow the definition of d-dimensional manifold as being locally homeomorphic to Rd Lee 2012), meaning that the dimension of manifold does not vary within the manifold. M will denote, depending on context, either embedded d-dimensional submanifold of TRD, or countable disjoint union of embedded submanifolds of potentially varying dimensions. We will assume that data of interest is supported on M and generated according to a probability distribution admitting density p : -M- - R Writing M as M = UyM, where Mj is dy-dimensional and M; n.M, = 0 for * j, the local intrinsic dimension of x E M is defined as the dimension of the manifold that x belongs to, i.c LID(x) = d, if x E M;
Note that p is not a density with respect to the Lebesgue measure RD. but rather with respect the Riemannian measure (or valume form if it exists) on M; see the work of Loaiza-Ganem Ganem al. (2024) for a discussion about this setup.
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We will make heavy use of isotropic Gaussian densities with log standard deviation 8 R, i.c. with covariance matrix TD; for a mean ja DR we will denote the corresponding density evaluated at IR
paur-pplie 6D( = Ch e D8 exp (-2110-- uIle 20) where CØ = =2x) D/2
-
(2)
and where - II will always refer to the Euclidean norm. We will sometimes find it convenient to write M,, where d < D and z. p RD with the understanding that
Na(z; p, 6) = Co ed exp (-211z 1l122-22
(3)
is not density as it does not integrate to 1. We will also heavily use the convolution of p against a zero-centred Gaussian with log standard deviation ö, which we denote a
ev(x, 6) = J. p(x')No( - -0;;0,';00:: 8,,)r''
(4)
We will treat uniform densities analogously to Gaussian ones, and will denote the density of uniform random variable on a ball of log radius centered at μ R,, evaluated at 2, as
Up(æ; 6) = ChS 1x E BD((,e")), where = D (2 +1)
(5)
and where 1() denotes an indicator function, Bp(u,r) r) denoles a D-dimensional ball of radius r centred at n, i.e. rp( {a ED: || xlll r}, and I' is the gamma function. analogy the Gaussian case, will also use
M,, = 10e-11((= Bp(u,e"))
(6)
for 1, E RRD,, and will also write
(x) = / p(a') - 2';00
(7)
As mentioned earlier, we will also prove Equation when ex 6) is replaced by eu(x, Since
ex 8) JximFotade, p( pla')cye-Doo Che DĀ Pxpxpp (X E BD(x,ee)),
(8)
our result thus explicitly links LID(x) to the probability assigned by p to ball of log radius 5 around æ through
LID(x) lim 8-
log Px~p(X BD(x,e²)).
(9)
a6
2.2 Diffusion Models and FLIPD
As already mentioned, the main goal of our paper is to prove Equation 1 under general and realistic assump- tions. In the rest of this section we cover the background that makes this equation relevant, namely DMs and FLIPD. Our proofs do not rely on the content presented in this section, which can safely be skipped by readers who are already familiar with these topics, or by those who are only interested in our theoretical results. We will follow the stochastic diffcrential cquation (SDE) formulation of DMs of (Snn et al. (2021) DMs are generative models whose goal is to learn p, they do this by first instantiating (forward) noising process through an Itō SDE,
dX, == a(X,, t)dt + 3(t)dW, Xo ~pp
(10)
where a R x 0, + R and B [0, 1] + R are fixed functions which are specified as hyperparameters. and W, is a D-dimensional Brownian motion. We will denote the density of X, implied by Equation 10
Note that, since p is not deusity with respect the Lebesgue measure, the integral in Equation 4 is not Lebesgue integral, and dx' should understood as the Riemannian measure or volume form on M. Nonetheless, (N(.,)) is Lebesgue density in RD due the added Gaussian noise.
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p(,,t) .t; and note that 00) _ Here E [0, 1] corresponds to artificial time variable specifying how much noise has been added to the data through the SDE, with ( t= 0 corresponding to no noise, and with t = 1 corresponding to p(- being "almost pure noise". A surprising result by Anderson (1982) and
Haussmann & Pardoux 1986) shows that the reverse process, Y, = X1-1, also obeys a (backward) SDE,
dy ()__-))s(YY.11 a 1-1,111110 1t)] dt + B(1 -t)dW, Yo p(.,
(11)
where 8 is the (Stein) score function, i.e. s(x, 1) V logp(æ; (), and W, is another D-dimensional Brownian motion. [Equatinn 11 is at the core of DMs: by approximating the score function with a properly trained neural network & (and also approximating p( 1) with an appropriately chosen Gaussian distribution). numerically solving the resulting SDE until time == 1 will produce samples from a DM these samples will be approximately distributed according to p; the better the approximations made throughout, the closer the distribution of these samples will be to p (De Bortoli 2022).
The function a is often chosen as a(x, t) y(t)x for some function y: : + R. Under this choice, the transition kernel corresponding to Equation 10 is known (Särkkä & Solin 2019), and it is given by
P1p(2 mb) = (z (t)xz, log at)):
(12)
where the functions vi, a [0 1] + R are uniquely determined by Y and 3, and can be numerically evaluated for all common choices of and B. Furthermore, under these standard choices the ratio X(t) == att)/ (t) is injective and therefore admits a left inverse X-1, which can also be nunerically evaluated. Recall that the defining property of the transition kernel is that it satisfies
p(z, t) = fxx plz' Pejoo (z) aa)dd'.
(13)
The density in Equation 13 resulting from the SDE in Equation 10 is extremely similar to the convolution in Equation both of them operate by adding Gaussian noise to data from p (up to the rescaling of the mean by (t) in [Equation 12). The main difference between these two ways of adding noise is simply how the amount of added noise is measured: in the DM formulation, no noise corresponds to t O, whereas this setting corresponds to the limit as 8 + -00 for convolutions. Intuitively, it should then be the case that if one has access to a DM, then the right hand side of Equation I can be approximated, thus obtaining an estimate of LID. Kamkari et al.] (2024b) used this intuition to propose FLIPD. First, they showed that ev(., and p(., .t) are related through
log @x( Do (lo))(((())) + log (((()))x,,())))
(14)
where ((0)= == (e"). Then, using the Fokker-Planck equation which provides an explicit formula for oe ((x,t)) t) along with the chain rule and Equation 14 Kamkari et al. (2024b) showed that
i log (x - 0² (1(0)) Tr (V=<(((1))), + ([((v(())= 08
-(,,,),))))) (15)
Importantly, evaluating v requires access to p(x, f) only through 8. Lastly, by using the trained score function to approximate the unknown and by setting a negative enough do, FLIPD is obtained by combining
Equation 15 with Equation I as
FLIPD (z; 5o) = D + (88 x, t(66))
(16)
The correctness of FLIPD as estimator of LID therefore hinges on Equation 1 holding in general, and previously mentioned, Kamkari et al. (022bb) only proved Equation in the case where M is affine submanifold of RDD We refer the reader to Kamkari et al. (2024b) for more thorough derivation of FLIPD, along with illustrative examples, and empirical results.
Note that p(..0 and .1) for not densities the same sense; only the latter are Lehesgue densities.
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3
Related Work
As mentioned in the introduction, statistical estimators of LID typically rely on the computation of all
pairwise distances or angles on given dataset sampled from (Fukunaga & Olsen 1971 Pettis et al. 1979:
Grassberger& & Procaccia, 1983 Levina & Bickel, 2004 MacKay & Ghahramani] 2005 Johnsson et al. 2014
Facco et al. 2017 Bac et al. 2021), resulting in these estimators exhibiting poor scaling on dataset size and ambient dimension D. More related to our work is research aiming to leverage deep generative models for LID estimation. SSanczuk al. (2024) proposed an estimator which also leverages DMs but is not based on Equation and requires many more function evaluations of & than FLIPD. Horvat & Pfister (024)) proposed another method using DMs for LID estimation, but it requires altering the training procedure of the DM. As result, these methods are not compatible with off- tff-the-hheff state-of-the-an DMs such as Stable Diffusion
[Rombach et al. 2022) to the best of our knowledge FLIPD is the only estimator which scales to this sctting, making it particularly relovant to properly justify it.
Other works have used generative models beyond DMs, for example Zheng et al. (2022) showed that the number of active latent dimensions in variational autoencoders (Kingma & Welling 2014 Rezende et al,
2014) estimates LID. Lastly, Tempezyk et al. 20022] proved another convolution-related result which they
leveraged for LID estimation with normalizing flows (Dinh el al. 2017 Durkan el al. 2019) More specifically, they showed that for x E M, as 8 + -00;
ex 0) O(LID(z) - D) + 0(1).
(17)
By adding different. levels of noise 5 to data and training a normalizing flow for each noise level, Temppzyk et al. (202)) estimate log ex(xx) 5) for various values of 8; they then fit linear regression predicting these estimates from 8 the resulting slope is an estimate LID(x) D thanks to Equation 17. When proposing FLIPD, Kamkari et al. (2024b) noted that Equation 17 provides informal intuition for why Eqqation 1 holds: if the (1) term was constant with respect to 8, then Equation 17 would imply Equation 1 In this sense, proving Equation boils down to showing that the (1) term indeed behaves like a constant as -00, or more formally, that the limit of its derivative with respect to converges to 0.
Several works have studied the interplay between DMs and the manifold hypothesis in contexts beyond LID estimation: Pidstrigach (0022) first showed that if the error between & and is bounded, then DMs recover the correct support, thus showing that DMs can learn manifolds. De Bortoli (2022) refined this result with error bounds in Wasserstein distance, and various follow-up works have provided finite-sample estimation rates (Chen et al. 2023 Oko et al. 2023: Tang & Yang 2024 Wang et al. 2024). Lastly, Chen et al. (2024) observed that, under the manifold hepothesis, the Jacobian of the score function has low rank, and they further leveraged this observation for image editing.
4 Result Statements and Discussion
In this section we state and discuss our results. Partial proofs that avoid differential geometry (Lee 2012
2018) are provided in Section 5) the remaining steps, which rely on these tools, are given in the appendices. In first. result, we establish that [Equation 1 holds when M is arbitrary submanifold of RD and p obeys continuity and finite second moment conditions.
Theorem Let M be a smooth d-dimensional embedded submanifold RR and let p be a probability density function on M. Let E M be such that p is continuous at z, p(z) > O, and C I==J=- p(x") p(x')distt distty (z, xf d''d'' < 0o, where distM denotes the geodesic distance M obtained from the induced Riemannian metric M. Then,
θ log (x( d D.
lim oo a8
(18)
Our continuity and second moment assumptions in Theorem 1 are mild and are completely analogous to corresponding assumptions made by Kamkari et al. (2024b). Nevertheless, we emphasize once again that Kamkari al. (2024b) also assumed that M is an affine submanifold of R which is a much stronger and unrealistic assumption on the structure of M. Watchful readers might have noticed that the finite second
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moment assumption - ie. SM (I,, oc implics that M must be connected (or more precisely, p must assign all its probability mass to a single connected component of M), which is likely not realistic in many settings of practical interest. However this obstacle is minor since, as we show below, Theorem can be straightforwardy extended to the case where M is a disjoint union of submanifolds of I of potentially varying dimensions.
Corollary 1. Lel M = U,Mj where M2 is a smooth dy-dimensional embedded submanifold of RDD with = minizy inf,, EM,,,,,, ||x;- zyll O. Let p be a probability density function on M, which we write as p(a) = [)))(z) for x MM, where for every J. Pj is a probability density Mj and πjj 0, and E2 -; Let x E M₁ for some be such that Pt is continuous at æ, pi(x) > O, and JM. pr(xa) (z, x' a')da' < 0o, where distm, denotes the geodesic distance on M, obtained from the induced Riemannian metric on M, Then,
a log ex ex 6) = d, D.
lim 6- do
(19)
Corollary formally justifies FLIPD the sense that it shows that, under mild regularity conditions,
lim D u(s t(o) LID(x),
(20)
that is, when the score function is perfectly learned, i.e. & 8, have that FLIPD (1; - LID(x)
8 + -00.
For the uniform case, analogously to the Gaussian one, we first prove the case where M is d-dimensional submanifold, and then extend the result to disjoint unions of submanifolds. Note that our results with uniform distributions do not require the second moment conditions required the Gaussian case.
Theorem 2. Let M a smooth d-dimensional embedded submanifold of DR and let p be probability density function on M. Let x E M be such that p is continuous at x and p(x) > O. Then,
lim 6-2- 0o 08
log (u(xx, 6) = D.
(21)
Corollary 2. Let M U,M where Mj is smooth d, dimiinnnii embedded submanifold of R with Ç == minizj inf ini,, [| 2y > 0. Let p be a probability densily function on M, which we wrile
p(x) = (x) for x E Mi, where for every j, Pj is probability density on Mj and Tzz > 0, and E, n3 = 1. Let x E M; be such that Pe is continuous at x and (x > 0. Then,
lim 8-
log ex, 6) D. 06
(22)
As previously discussed, Corollary immediately links LID(x) and (X= Bp(x, through Equation
Albeit [Corollary T] is more relevant due to its connection to DMs and FLIPD, Corollary 2 still succeeds connecting two quantities of interest in machine learning despite lacking these connections. Some older estimators of LID use a given dataset sampled from p by regressing the log proportion of dalapoints in
Bp(z, e²) against 6, and then estimating LID(x) as the corresponding slope (Pettis et al. 1979 Grassberger & Procaccia 1983). These works provided an intuitively correct but not rigorous mathematical justification behind this estimator, which Equation 9 rigorously justifies.
5 Proofs
5.1
Gaussian Case: Theorem 1 and [orollary
Proof of Theorem 1. The core idea behind the proof of Theorem exploit the following relationship between (1, and Na(2; (1;;;;;;))
Nn - x': 0, = sld- ( Na(
(23)
Note that, because & >> 0, without loss of generality any density p on can written as p(r)) (2))(2)) when E Mi-
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Thus we have
()(2,0)= = J.a
- ( = (2x) A-D 2 ((ld-)) J,
pla')Naax (2;0,)dxt
(24)
M
To simplify notation, we let
(x) = 1SA p(z')Na(x - d';0,5)dz',
(25)
so that
0 06
log ex 8) = d - D d
log p (x).
(26)
08
Thus it suffices to show that First, note that have
0 log / (x) 0.
lim 6--c0
(27)
06
2o%a Na(a - Nalx- x':0,,0 = (-d || 26 2':0.6).
(28)
06
Then,
lim
i
N (x) log ps
lim
06 18 (x) ps (x)
(29)
06
6-2-
2b J pla Wa p(a)) x' 00, 8).)dr
lim
(30)
SMP(I') Nalz - 02; 8)dx'
IMP(a') o Nal2 z²: O, 8)
lim
(31)
6--000
Anpla")aaaa d';0, 0,d
SMp(aa) d + 11 220 -x';
x'; 8)da'
(32)
6--00 00
SMP(e')Nolz - 2²;0, SMP((') N'113Nal - x'; 5)dd'
lim
(33)
8---0
IMP((')Nall - x': 0, 8)rr'
Note that going from Equation 30 to Equation 31 involves moving the derivative inside the integral. We
formally justify this step in Appendix A We then leverage the following two propositions, whose proofs we also include in Appendix A
Proposition 1. Under the assumptions of Theorem 1
lim 8- M
p(z')N((x - -2:; = =p(x).
(34)
Note that Proposition I is very similar to the limit of Gaussian being a delta function, i.c. for function f Rd R and y Rd we have that Sge [(y')aa(y 0, 8)dy'' f(y) d -c0. However, the differences here are that: (i) we computing the convolution manifold M, and (ii) the distance in the "Gaussian" is the distance of the ambicnt space, RD..
Proposition 2. Under the assumplions of Theorem 1
lim e e-2 S p(z')ll= o' [ (1;10-2;00,0)dd - d p())) 8--00
(35)
The expected squared norm of centred Gaussian is well-known to be the trace of its covariance, i.c. SEa lly y Na 0)dy/' = 26 Intuitively, Proposition 2 can be understood as stating that this property still holds as + -oo under (i) and (Mi), with p(x') again leaving the integral as p(x) because the Gaussian behaves like a delta function in this regime.
Applying these two propositions to Equation 33 yields Equation 27 which in turn finishes the proof of Theorem

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Proof of Corollary 1 Under the conditions of Corollary
ex 6) is given by
ex(x, 6) - Ti J.n py(a') - o';
(36)
Following an analogous derivation to the one leading to Equation 33 we gel
Eg Ty Jm, Pj ()()') - 24 MND( o':0,))dr'
① log (x) - DD + lim 8-2-00
lim 0c 06
(37)
Σ3 ms JM, P5(z')N(( - 0,6,d
Using basic calculus to maximize (D - x'; ; 0, 6) and || - e 25 -2;;,, with respect to || - x'1
subject to || x'|l 2 yields that, when this constraint holds,
No N';0-=';0,0) 0) S Che D8 exp( exp(-1152 _ 2
(38)
and that if additionally
S 3 log S. then
2 x' e' 1eleee=2NNee 10;0==';006) S -2 D2 exp(p _ 22).
(39)
The bound in Equation 38 decreases monotonically to 0 as ö -o, and the bound in Equation 39 does so as well provided that 8 < log - 1 log(1) + 2) (which makes the derivative of the log of the bound with respect to positive). It follows that ND(x -
and || 2' 10--2M40 2ND(2 - xz; (0,6) are both unffrmmy bounded over x' and 5 when x' E M, for j + and 8 < min(? log logg - 2 log( ( + 2)). The dominated convergence thcorem then guarantecs that, for j + i:
lim Im M,, 8- -0
- Ps('' - -';0,,)d'' = =0 lim Ju, p;(x')|x=- Ile 12ee-20 26 NN(x - -1;0, o)dz'.
(40)
5--000
Lastly, since a E Mi, tion 37 reduces to
Proposition
ensures that lims or SM. Pil 2'; 8)dx's > 0, so that Equa-
SM. pr(x')lz= - 2'1ee 28 No z': 0)d
lim
log ex(xx ()-18))=-)) =- D lim
-
(41)
06
PLE'(NN) x': 0, 0,8)dr
i
lim
log
Pi(e'NN(( o';0,0)d'' d, Dr
(42)
0a d6
M,
where the last equality follows from Theorem
5.2
Uniform Case: Theorem 2 and Corollary 2
Proof of Theorem (informal). In analogy with Theorem the proof of Theorem 2 hinges on the following relationship between Up p, 6) and (a
Un(z;n,8) (C(())))))((())) o(d 1eddd))uux
(43)
Thus we have
cul21 6) = I p(=))up(s-x''000) -BCC ((dddDD) 1 plal)Ml(xx - 21;0,8)ok
(44)
To simplify notation once again, we let
(x) = S( p(z))Ud(z (0,))''.
(45)
so that
0 log ou (z, 8) = d D
log ps (z).
(46)
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Thus it suffices to show that
lim log (x) 6- 06
(x) O.
(47)
Despite seeming extremely similar to the Gaussian case from Equation 27 the proof of Equation 47 is not completely analogous. We now provide some intuition as to why this equation holds. Note that P/ (x) = Px E BD(xzee)). Intuitively, since p continuous at æ can be locally approximated by a constant, meaning that when + -oo, we should expect Px~p(X e Bp(x, e")) to behave like a constant times the volume (in M) of Mn BD(,,e". Since M is d-dimensional, we should also expect MnBpla e²) to behave d-dimensional ball as 8 -00, ic. to have volume proportional to ca5 Putting these intuitions together, get that pf((1) (x) behaves like constant as ō -c, thus yielding Equation 47
We formalize the above intuition in Appendix B by providing a rigorous proof of Equation 47 completes the proof of Theorem 2
this result
Proof Corollary 2. Under the assumptions of Corollary 2 @ur(2; 6) is given by
eu(x, (01x;))==m1 Tj S M,
(48)
Whenever < lo if a' E M, for j i we have that Up(x - x'; since a E M, in which case
S SM. ((('')u(x=='''
@u(2, ni
(49)
By Theorem 2 it follows that
a log I val'")n((=-<''00 d; D.
lim oc 06
log en (2) lim
oo oo 06 J..
(50)
6 Conclusions and Future Work
In this paper we established a formal link between LID and the rate of change of the densities of different convolutions. The Gaussian case is of particular interest as it provides formal justification for FLIPD, a state-of-the-art LID estimator. The uniform case provides an interesting connection between LID and the probability assigned to Euclidean balls, and rigorously justifies older LID estimators.
One avenue for future work would be extending our result beyond Gaussian and uniform distributions. For example, despite losing direct relevance in machine learning, finding sufficient conditions the convolving distributions for our result to hold is interesting mathematical problem. In particular, that, under adequate regularity conditions, Theorem and Theorem 2 can be generalized to cases where the noise distribution that is convolved against obeys a decomposition like those of Equation 23 23 or Equation 43)
hypothesize
Another interesting avenue for future work is extending LID estimation to flow matching methods (Lipman et al. 2023 Albergo Vanden- Vanden-Eijudenl 22023 Liu et al. 2023 Tong et al.] 2024). These methods aim to train vector field 0 such that solving the ordinary differential equation
dY, 6(Y,, t)dt, Yo T0~p(-,11 p(,,1)
(51)
results in Y1 being distributed according to the data distribution p(,,0). Flow matching methods differ in how they obtain such 0. For example, when using the so called lincar interpolation method (Lipman et al. 2023), 0 is equivalent to an affine transformation of a score function & (see Appendix D.3 in Kingma & Gao (2023)), which means that FLIPD can be trivially applied in this setting. However, not all flow matching mothods produce á from which one can casily recover a corresponding score function plug into FLIPD: for example we should expect this to be the case when using several rounds of rectified flows (Liu et al., 2023) optimal transport regularization (Tong al.] 2024 We believe that producing tractable estimate of
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Published in Transactions on Machine Learning Research (10/2025)
(ID(z) while given access only to successfully trained è from any flow matching method is interesting research problem.
Lastly, although our results indeed justify FLIPD, they do so in the setting where the score function is perfectly learned, i.e. $ = Despite DMs being highly performant, they never perfectly recover the true score function, and discrepancies between & and 8 can result in
lim FLIPD (z; 6) = LID(x) 60-20
(52)
being violated. Indeed, Kamkari et al. 2024b) reported that in some instances, FLIPD can produce negative estimates of L.ID. Since Kamkari et al. (2024b) only proved [qquation 1 when M is an affine submanifold of IR the possibility remained that this negativity was due to the result not holding more generally; a consequence of [Crolllary iss that negative estimates can only be caused by having imperfectly learned We believe that quantilying how much FLIPD (æ; do) differs from LID(x) as function of the error incurred by & to approximate is another relevant problem. In particular, note that if we had bounds on both the error between 8 and 8, and on the error between Vs and V8, as 6 + -c, we could trivially obtain a bound on [M(s, x, f()) - v18, xr t(8)] (see Eqquaioo 15); in turn, this would provide a bound on the LID estimation error. While we are aware learning theory work bounding the error between and å (Chen al. 2023 Oko al. 2023 Tang & Yang 2024; Wang et al. 2024), we are unaware of any existing bound on the error between and V8; finding such bound is her interesting direction for future work.
Acknowledgments
We thank Hamidreza Kamkari for useful conversations during the early stages of this research.
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A Proofs for the Gaussian Case
The following claim provides justification for taking the derivative inside the integral from Equation 30 to
Equation 31
Claim 1. Under the assumptions of Theorem 1
á ((a')Na(x - =1;00 S p(x') 8 Na x';0, 00),)'''.
/
(53)
06 M
06
Proof. Let 8 R, and let & be such that da < 8. We begin by bounding 2 NaNa(I 0,;
Na(a 2';0,8)1 d + || e'112e-2 -26 1 exp (-11) | x' x1112t-
(54)
00
(d || x' el - 2h CO e- dba.
(55)
This bound is integrable duc to the finite second moment assumption:
J pld')(d ||x - s' r'1 11fe-B80 26n C6 e- dd s / pla') dist (i, 20, -d5 <0.
(56)
The result then follows from the Leibniz integral rule.
In the rest of this section we prove Proposition and Proposition 2 the rest of the section, i.e. some of the notation introduce in the proof Proposiiio 1 is also used after this proof.
Note that we share notation through
Proof of Proposition
Recall that we want to prove that
lim / pla' Na(e- - =';0,,)d'''' p(x).
(57)
To compute this limit, without loss of generality, we perform translation so that x 0. Note that this translation preserves distance. Then perform an orthogonal transformation on ReD such that the tangent plane of M at x = is the d-subspace spanned by the first coordinate vectors. Denote these coordinates by (y1,.... ya). As M is smooth, M can be parametrized by {y. u(y)} near x 0, where y = (./...., 9.) and u is function from some open U C Rd to R-d,, such that u(0) 0 and (Du) o In other words, (yı,-.., Wa) local coordinate system of M U, with the induced Riemannian metric g- Note that g = la at as (Du)(O) 0. This means that U is diffeomorphic to open set in M via o(x) (a, u(x)). Shrink U if necessary such that U is bounded, i.c. R = (d(U) ||a'|| < cc. Further shrink U if necessary such that (0, R) n.M = ((U). In other words, x' E M, | < R + x' E &(U).
We are going to compute the limit on some open V C @UU) in M and MY V. Let Ry == supz'ev ||-0|| < <0. Then for x' E EM\VV, for any a E R, we have
a5 Nd((='';0.) = (2T) 4/2 8(d-a) exp
1 1//
(58)
2
1 R& -2
- (2T) d/2 5(d-a) exp
-
(59)
2
which approaches zero as 8 + -00 This means that for any e > 0, there exists KV,a such that 8 < Kva implics ea Na Na(-=':0,0) <c
3oo avoid using non-standard notation for derivatives, overload notation and refer to both the derivative operator and the ambient dimension. The meaning of will always be clear from context.
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If a/ E V, in the coordinate system (y1.....31a yaa), we have Na(-=z'0,,) represented by
(v, = (2x) d/2 dd exp
1
(1uu)]++
26 20)
(60)
For simplicity, let
5d exp (-SlbolPe - )
2/2e 6) (2x) d/22- 2-d2e-6ee
(61)
Note that (y, 6) is the standard multivariate gaussian distribution with covariance e22 la-
It clear that Ñ (y. 6) S Ny, 8). Now, consider the function v(y) = Iu(x]], Note that v(y) is continuous
MM
everywhere in U \\ [0). Since and the derivatives of u vanish at 0, from the definition of derivative, we have
lim (u(y) u(0) (D)) lim |
lim u(y).
Myl
Thus v(y) can be extended continuous function U.
Now let Kv max v(y)- Then we have [|u(y)]) s K& |1ll2. Thus (VV)
(g. 6) = (2x)-aa/2 - exp
2 (1al1 + [|u(()) (1)) -22)
(62)
2 (2x) 4/2 6d exp
2 ||al12(1 + KeZ)e -2
)-26
(63)
e (2TT) - d/2
(6+6o)d
exp
-1ll11= (11111(2-2(6++60)
(64)
(1+K3) 4/2 (2,, +
(65)
where & = = log(1 + KZ)). To summarize, we have
(1 +
S (v,,,)) NN,,0)..
(66)
Now, for any open V c ((U) in M, we have
ps(0) I p(z') p(a":,"(a
(67)
M
I p(a' Na'')a(-='',,,) + Nal'"Naa-
(68)
For any > whenever 8 Kvo have,
JAY plo))Na(=d':0,))ddr S JaY plze)eda' se plad)do' = c,
(69)
p is probability density M. This shows that
lim 8-100
JAY ('=')Na(==''0,,)) = 0.
(70)
Note that this is true for any open V C (UU) in M.
On the other hand, in the local coordinate system (y1,,..,,ya) Wy), we have
((lo))a((=2:0000 - l(v) p()) ol))
(71)
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Published in Transactions on Machine Learning Research (10/2025)
Here g(y) (et(gs)(v)) is continuous (V) with g(0) 1. From Equation 66 we have
(1 + KV) 4/2
( bo)dy
(72)
6-
p(v) (w))g(y)NN 9 (y, 8)y
(73)
(0)
[ (--10)) ply) (y))
(74)
Since N (y, 8) is the usual multivariate Gaussian distribution, converges to the delta function. Thus, taking -o, we have
(- Plu) & lim / 8--00 (-1(
4/2/2)/--(p(000 S lim inf
p(y) Voll S p(0), (75)
since v //(0) = 1. But we have
lim inf p5 (0) = lim inf
p(a))Na((='';0.)
(76)
6--30
8-0-00
M
J, (('')Nal->''s 8)dx'
lim
0)d) + lim inf 8---00 p(a')No(-2' 8)dz
(77)
6--00
Me
lim inf
(78)
in
p(u) V g(v) (v. 8)dy,
(79)
1(V)
and similarly for lim This mcans that
(1+ (1+ 400ppp))pppppppppppp))))p))))4 < lim inf p N (0) I lim sup p (0) p(0).
(80)
6-
Note that these inequalities hold for all V C ((U). First notice that the function f(x) = (1+ + + is continuous at = 0, that f(0) = 1, and that f(z) < 1. For any e > 0, these exists k > 0 such that whenever |>| < Ks we have - (1 + 22) <e As v(y) is continuous al y = 0 and v(0) = O, there exists > 0 such that whenever ||y| k', we have v(y) < x/2. Thus let V = (BBa x') nU), we have Ky S k/2 K and thus (1 K3 d/2 1 Thus have
p(0) - e) liminn 1 00
(()) (0) lim v < p(0)
(81)
for every 0. This shows that
lim pe (0) p(0).
(82)
8-
Before proving Proposition 2 need to establish some basic tools.
Claim 2. Let Ba(O,r) be the closed ball in IRd centred at of radius r. Then
lim Bd (a(.))
-- N(y, 0.
25
(83)
Proof. Without loss of generality, assume < logr - log (d + 2), then the integrand will decrease as decrcases (the derivative of the log of the integrand with respect to 8 is positive) and converges pointwise to It follows that the integrand is bounded for negative enough and the result follows from the dominated convergence theorem.
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Claim 3. Let W be open and bounded subset of R with 0 W. Let f W R continuous ot 0 and bounded on W. Then we have
lim [ f(v)llul?e-2 N(y, N(y f(0)d. 6-
(84)
Proof. For simplicity, let (y. 8) 1lylee 28 Ny, 8). First we note that Ny, 6) the donsity of multivariate Gaussian distribution with covariance matrix 206 la. This means that.
fet
|lv1 Rd
[|lllll Tr(c2* de
(85)
an As f continuous at O, for any <> O, there cxists r > 0 such that lyll < (f(y) f(0) <c Let K be
upper bound of in W. Using Claim 2 can pick K' such that when d < K' we have Sg-1 Ba(orr) Hy, Then for 8 < K', we have
f f(v) 5)dy f(0)d
(86)
[ f(v) 5)dy [ ((0) 8)dy
(87)
(f(y) f(0)) +
Hl.) 8)dy
((0)H
(88)
Ba(O.r)
JW(Ba(Orr)
Hly 8)dy + K / JW
<6
H o)dy + [f())]
Hy. o'dy
(89)
Ba(O.r)
$6 Jgs Hyyd8)dy+ K
H(u + [(()
Hu,,
(90)
[R(.B.(0rr)
Qu (2(,r)
<de + Ke [f(0]]c
(91)
=e(d K + (f(0)).
(92)
This proves the claim.
0
Proof of Proposition Recall that we want to prove that
lim e e-28 S d/ 8--00
= d. p(x).
(93)
We perform the same translation and orthogonal transformation as in the proof of Proposition 1 which ensures that x = 0 and that the tangent plane of M at a = 0 is the d-subspace spanned by the first d coordinate vectors. We once again consider an open V C ((U) in M such that 0 E V, and break the integral on and on M \ V.
For any > whenever 8 Kv.-2 we have,
S.Y p(z (('')lz'||ee 26 Naal='';0 S / p(z')]la'1
(94)
MiV
Se I May p( p(za)d )distiv (2', O)dz'
(95)
Se / p(x') p(r')distz (x', 0)dx'
(96)
MM
=Ce
(97)
Here we used the assumption that the expected squared distance from x 0 is finite. This shows that
lim I Jn
p(a'))1='11 22 0) 0.
(98)
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Published in Transactions on Machine Learning Research (10/2025)
Note that this is true for any open V p(U) containing z 0 in M. On the other hand, in the local coordinate system we have
| plad))ld1PPN(a-d000
[0-(()) ( + I1(()))))) 5)dy.
(99)
By Claim 3
we have that
26
p(v) v(v) [c/) o)dy
(100)
(V)
= 26
p( p(u) /1 Ñ (y. dydy
(101)
(-1(M)
Se 26
p(v) pwW 8)dy
(102)
(-)
-p(0) V 9(0) -p(0)
(103)
(104)
as - -00 Here use the fact that f(v) = ol is bounded, because o= (V' is compact. The lower bound will be
6 28
p(U) PU0////0 d)dy
(105)
(v)
28
p (tl.://lll((++0000)00 5)dy
(106)
(-((V)
2(1 + K2)-4/26
1/2
p(y)
(1) (1+ do)
(107)
(0-1())
(11 + (2))k////1111
4/2 (0)1
(108)
=(1+ -11+33)
(109)
as - oo. As in the proof of Proposition we can bound the limit inferior and limit superior and shrink V as we like, thus finishing the proof.
B Proofs for the Uniform Case
Proof of Theorem 2 Recall that is enough show that
i 08
lim log pl (z) 0.
(110)
We begin by performing the same rotation and translation as in Appendix A and considering the same coordinate vectors, parameterization of M, and open set U. Note that we have
(0) 1 p(x (aa')Ma(='';0,,)d = cd Ju pla'dd''.
(111)
Assume that nogative enough for M MnBBp(ece C (UU), which is open in M. Thus, in the coordinate system (Y1,..., ya), we have
-d8 (0) = cy e d8 S p(y) p(u))olu)ds
(112)
19

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Published in Transactions on Machine Learning Research (10/2025)
where V/3 == MM Bp(x,e")) {v: |lvll + |u(1))1 2² glv) det(gis (v)) is continuous Va, and ((0) = 1. Now, we have
a
1
a
log (0)=
(
(113)
06
(0)
06
1
dpp (0) + cye 2
08 /,, (lu)/olyi)
(114)
(0)
ply) V /y
06
= =dd
(115)
p(u) V /luidy
So, we must prove that
d6 / ply) Vilu)dy
lim 6 00
V/
(116)
p(y) (ly)vg(vi) slyvdy
VVa
Now, let As = 0 Sv. dy be the derivative of Vol(V3) with respect to 6. Since PVD is continuous at y = 0,
58
we have
/ p(y) (ly)a
1 (f
ply)Voludee [ plo)
lim h-20 h
(117)
05 v/
Veyn
lim 1
{
plu)VV(u)dd)
(118)
h-0 h Vs.n
2((+1))) < case where is analogous. Note that for small enough e2 and h, Vs, diffeomorphic to a d-dimensional annulus, U is a local chart for ((U), and for small enough open set, the intersection of the ball in R with M is diffeomorphic to the ball in Rd.
where V&. {y: 26 < ||ul)+||u(y)]1+++<
In what follows we will consider the case where h > 0: the
Now for any e > 0, there exists & > O such that (p(y) / g(y) - p(0) < whenever ||y|| < K,, since g(0) = 1. Choose K' such that when < K' we have Vo, C Ba(0, Note that a priori K' depends on h. But since we are interested the behaviour as + O, we can choose K' so that the condition holds for h ho for some hg O. This means that for 5 K', we have
L (p(u) 000) p(O)ddy Ve.h
I h /// 1./. p(v) - p(O) < E f dy.
(119)
Va.h
VV.
This means that
p(0) +c { dy. p(0) - / dy < 1 / Plu)ooluddy
(120)
h Van
van
h vvan
Taking h + O, we have
(p(0) c)As S [ p(y) Voy)dl S (p(0) e)As. 06 Jv,
②
(121)
Note that similar result above for the same k and K' applies the denominator of Equation 116 i.e. without the derivatives,
(p(0) (oooVV) I (lu))olr)d < (p(0) (V)((V))
(122) together, we get that, for 5 < K":
Putting Equation 121 and Equation 122
θ
p(u) Mu)Votaa V(u)doluldy glufdy
(p(0) - A) (p(0) + (oo((V))
a6
(p(0) + +)
V.
(123)
(oll))
V/
20

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Published in Transactions on Machine Learning Research (10/2025)
To calculate As, use the Leibniz integral rule in its most general form Flandders 1973
/ dy Ve
/ Vvs
'sddy + S 4dy + / V/
dy= = / 4,dy,
As
(124)
06
Jov.
as dy is closed and independent of 6. Here is denotes the interior product with i, which is the vector field of the velocity, i.c. if & is a family of diffeormorphisms from a fixed domain V to Va, then & the derivative of 2s with respect to 8.
Thus, in what follows construct family of diffeomorphisms from a fixed domain to Va that then apply the Leibniz integral rule. We first establish some ground work for Vg. Note that Va {v G(v) < <)), where G(y) = 1 log([lv| + [[(())]²) Then we have
aG
3i + E, 4yys
(125)
Oy:
1 + ||1|
Oug where we use the notation that any indices after a comma indicate a derivative, ie. uj, Zy.
This means
+ (Du)u
VG
(126)
| ||]|2
Proposition 3. There erists an open set W containing such that VG is well-defined and vanisinig on W (0).
To show this proposition, first we establish the following facts:
Claim 4. Let f: IR R be smooth such that ʃ(0) 0. Then f(v) =, Vigi(y) for some smooth functions 9s.
Proof. Note that
a σe s(y)dd S/ Σ
y of Dy. f(ty)dt.
f(v) - f(y) f(0)
(127)
Thus 0: = So af (tyyddt and 9i is smooth. By,
Corollary 3. With u(y) as defined above, we have u(y) Eis Acy(y)yan for some smooth functions Aij Rdd + D-d
Proof. Since u(0) = O, we have up IRd - R such that us (0) (0) = 0. Thus, by Claim 4 up (x) = E, Iki (v) for some smooth Okj. Since all the derivatives of vanish at 0, for any k, we have ( = (0, IRi (0) + Σj (0) 0 = (0). Thus, once again by Claim 4 (y) = E3 hug (y)yz for smooth hurz. Thus we have u4(y) E, gxx(y)y En, (1)))1)) Thus (Asj)k = hkis, which completes the proof.
Claim 5. The functions IJ (Du) and u (Du)u can be eztended to Oa functions on Vs, such that each function and their first derivatives vanish at y O.
""
Proof. Note that since u and l are smooth, the only point at which we have to prove the above functions are C¹ is at y O. Using Einstcin summation notation, first notice that since u, AsMive3 from Corollary 3 have
Uj.i = AskI.UKe ASailt + AyaM,
(128)
which is a polynomial in of degree at least 1.
For u. u, we have
T a 4iu Aim Uk
(129)
where every term in this polynomial in y: is of degree least 4.
21

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Published in Transactions on Machine Learning Research (10/2025)
For y" (Du)u, we have
y (Du)u yu = Auurana (AgH,, + AJRiBl + Azazi),
(130)
where every term in this polynomial in y: is also of degree at least 4.
For u² (Du) (Du)u, we have
u u (Du) (IDu)((Du))) (Du)u 44, = (tm.
(131)
where again, every term in this polynomial in is of degree at least 4.
Let P(y) be a polynomial containing no terms of degree less than 4 (with coefficients being smooth functions on Vs), and Q(y) = P(y)/Ilyl?? It is easy to see that limy-o Q(y) = O. And thus Q(y) can extended continuously to y with Q(0) = 0. As P(y) contains no terms whose degree is less than 4, we also have limy-0 [Q()) ((v)-Q(0) 0. By the definition of differentiability, we have that Q is differentiable at 0 and Iell
VQ(0) O. Now, for + 0, we have aQ (v) Ty.
Tyy where P1() is polynomial such that every term is of degree at least 5. Thus aq 0 as 0. This shows that Q is C². Bys
Proof of Proposition This denominator of Equation 126 is |1yl + ||u]|2 = 11yl| + 5.t)
Since y
sly 0
as + 0, there is an open set around y = where + u * O. Thus for any y in this open set such that
y + 0, VG is well-defined.
The squared-norm of the numerator of Equation 126 is
Ily + (Du)ull= (y + (Du)u) Tw (Du)u) = uTy y (1 1 y y
2y (Du)u u (Du) (Du))(Du)u y y
(132)
Since, by Claim both 2y (DDuu and T (Duaj approach as y - 0, there an open set around y v' such that I (Du)ull² # 0 in this open set when * 0. This means that there exists open set W around = 0 such that VG is well-defined and non-vanishing on W \ (0).
As we are only interested in the behaviour of As when ò + -x,, we can choose a specific 8' small enough such that Vy C W. Now, for any 8 < 8', we have Vs C V8,, given by the sub-level sets of G. Now choose ba', Sas dg by such that 5a < 8a 8 < &, dy 8'. Now, define the vector field X v'(y) TVG|E [V6]) where * is a smooth function satisfying
{ :o
8, S G(y) S 6, G(y) 8a or G(y) > 50
(1y)
(133)
and S (yy)) < 1. Then X a smooth vector field that vanishes outside a compact sct, which implics that X is complete vector field. This means that integrating X gives rise to one-parameter group of differomophisms 6, Id - Rd such that 6: = X.
Now consider the map t + G()r())). We have
c(o,(v)) = VG dr(v) = VG- (v(((()) |
VC
= ((r(()))
(134)
0t
As (b((e(y)) is always between and 1, the map
+ G()r(y)) is a non-decreasing function of I.
Claim 6. 1ft200 0, then
G(y) < G(or(v) s G(y)+t +t.
(135)
Proof. We have
= f' d dr
G(O-(1))TT= = S' "(O-(1)))r..
G(o(())) G(o()))
(136)
Note that 5 & 1. Thus for 2 O, we have 0 < G(dr(v)) G(y) St
22

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Published in Transactions on Machine Learning Research (10/2025)
Claim 7. Let y be such that (ly) 5 E [6a. 6b]. Then for E [0,6, 6, 6), we have G(de(v)) G +t.
Proof. By Claim 6 if E [0, 6, ö], have G(dr(w)) [G(y), G(v) + c [6.. [o]. This means that
(v)((v)) = 1 for t E [0, 66 6]. Thus we can replace the inequality with equality:
G(or(w)) -G())) G(v)= = ((o-(w))dr = S/' dr =t.
(137)
Now, for any 8 [6), define 4 Q8-6.- Proposition 4. Ds is diffeomorphism from Vs. to Vg.
Proof. Let E Vo,, i.c G(v) < da- By virtuc of Claim 6 G(os(v)) = G(os- 8. (v)) I G(y) + 6a < 8. Thus (y( E Vg. Conversely, let y E Va and let y = Φ8, (y)) so that Ds(y') = It suffices to show that y' E Vo,: L.e. G(y') < da. Assume the contrary, G(y') 2 da Note that G(y') S because + G(or(v)) is non-decreasing. Then we have
G(v) = -(( (v')) ((())saw)(())) + G(y) 6
=
(138)
which is contradiction. Here used that l + G())(v)) is non-decreasing function for the inquullty and Claim for the second last equality.
Proposition 4 establishes that (0g 5 E 1o,, 5,]} is a family of diffeomorphisms from a fixed domain Va, to VG Vg. We can finally invoke the Leibniz integral rule from Equation 124 with = [aa on oV.. Note that
y + (Du)u
|| = 1ly + (Du)ul?
VG
VG
H()) (Du)u),
(139)
|y/| ||||2
(11/) 1|
VGI22
where (y) = +|
(Pujull? Note that of now, H (v) is defined on 0V, We would like to extend (y) to Vg. We can rewrite H(y) as
TT + (Du) (Du)a
y yy uytuu Hvy) y y 2y (Du)u + u (Du) (Du)u 1 1
V/
(140)
(Du)a vTv
3
(vy) can be extended to Va as C² function with H(O) 1. This means that can be extended to Va a C vector field.
Thanks to Claim 5
V
[VG
Now we apply the generalized Stokes' theorem, followed by the definition divergence, to get
A8 /
{ I ve dy = L de oa dy div VG llol2
Jov, dyy Jov,
||G||2 dy.
dy.
(141)
VG It suffices to calculate div T div H(v) (Du)u). Using Einstein summation notation, we have div H()) ( = (y) + ug,,,)), = H() uyauu, + ug,u + (1) + uy,uy)) (142)
which is continuous on Vg. At y 0, have
VG
div
(0) =0))d+0+0)+++ (1:0)(0++)
(143)
|
Thus for e >>>0 0, there exists such that if ||y|| < k', then (div TVG (r)) d <e. Choose K" such
that V8 c. (0,0, for K", Then
AAs - d Vol(V))) & J div |
VG
(2) d dy < <VVolVVs).
(144)
23

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Published in Transactions on Machine Learning Research (10/2025)
This means that
(d - v)Vo(VV) < As < (d+
(145)
Combining Equation 123 and Equation 145
we have
0
p(y) V
(p(0) - <) (c) p(0) +e
06
(p(0) + c) +
(146)
p(0)
p(y) g(v)dy
V/s
for d < K min(K', K"). Letting - O, we have 8 + -20, which entails Equation 116 and thus concludes the proof of Theorem 2.
24


Reviews:
[{"review_id": "ehHZ4ytQE8", "reviewer": "Reviewer_5wyd", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n1. General data assumptions and mild regularity conditions.\n2. Rigorous, well-structured proofs with clear logical flow.\n3. Consideration of non-Gaussian (uniform) noise, which is relevant in prior works such as Bansal et al. *Cold diffusion: Inverting arbitrary image transforms without noise.* (NeurIPS 2023) \n\n### Weaknesses\n1. No numerical validation or illustrative figures. \n2. Limited discussion of practical implications. For instance, practitioners might expect some characterization of the differences between Gaussian and uniform noise (given their distinct spectral behaviors), but the paper only establishes that both are valid for dimension estimation.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper aims to provide a theoretical justification for local intrinsic dimension (LID) estimation, clarifying the validity of recent diffusion-based dimension estimation techniques, such as FLIPD.\nThe main problem addressed is whether the derivative of the log of a Gaussian-noise–convolved probability density remains correlated with the local intrinsic dimension when the support is a general embedded submanifold (rather than an affine subspace, as in Eq. 1), and whether an analogous result holds when the data is convolved with uniform noise (Eq. 9).\n\nThe key technical approach for extending the analysis from affine subspaces to embedded submanifolds is a two-step strategy:\n1. Use the definition of a $d$-dimensional embedded submanifold to pull back the calculations to a $d$-dimensional Euclidean space.\n2. Apply appropriate bounding techniques in this reduced space.\n\nFor the Gaussian noise case, the authors compute the limiting probability (Prop. 1) and the expected squared norm (Prop. 2) for manifold-supported data. For the uniform noise case, the second step is adapted to bound the “area” of the annulus formed by intersecting the $d$-dimensional submanifold with two balls, reducing to a ring in the affine case. The proofs naturally extend to union of disjoint submanifolds, since the analysis considers a localized limit.\n\nOverall, the paper establishes mathematically rigorous results for problems that are intuitive but technically challenging in manifold dimension estimation.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "### Requested Changes\n\n1. It may be helpful to discuss related recent work that also examines low-dimensional or low-rank structures in diffusion models, for example:  \n   a. Wang et al., *Diffusion models learn low-dimensional distributions via subspace clustering* ([arXiv:2409.02426](https://arxiv.org/abs/2409.02426), 2024)  \n   b. Chen et al., *Exploring low-dimensional subspace in diffusion models for controllable image editing* (NeurIPS 2024)  \n\n2. While the main contributions are theoretical, adding a small numerical experiment or visualization could make the results more accessible to a broader audience and help illustrate the intuition behind the proofs (see Weakness 1).\n\n### Questions for the Authors\n1. The proofs proceed by isolating the $d$-dimensional term and showing the residual log derivative vanishes. Is this behavior specific to Gaussian and uniform noise, or might Theorems 1–2 extend to a broader class of noise distributions, e.g., sub-Gaussian?\n2. How should one interpret an estimated LID (or FLIPD value) that is non-integer? Does it indicate that the data only approximately lies on a manifold, or that the score function is imperfectly learned?\n3. Since the analysis takes the limit $\\delta \\to -\\infty$, equivalent to $t = \\lambda^{-1}(e^\\delta) \\to 0$, does this mean that only the score function $s(\\cdot,t)$ at $t=0$ is required for LID estimation?", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "IkdPHzIhCN", "reviewer": "Reviewer_RAVW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**1.**  The paper makes a significant theoretical contribution by rigorously proving the correctness of the FLIPD estimator for local intrinsic dimension (LID) when the data distribution is supported on a disjoint union of general smooth submanifolds. This removes the strong affine submanifold assumption from prior work and provides a much broader foundation for the estimator’s applicability.\n\n**2.** The paper extends the analysis from Gaussian to uniform convolutions, suggesting that the theoretical framework may be robust to the choice of smoothing kernel. This generalization could open up new directions for understanding the geometry of data distributions under different generative processes. \n\n**3.** The paper is well written and clearly organized, making technically involved results accessible to readers familiar with the field. The authors effectively communicate the motivation, main results, and implications of their work.\n\nPlease refer to the following section for main weaknesses.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper provides a theoretical foundation for the FLIPD estimator, which estimates the local intrinsic dimension (LID) of data modeled by diffusion models. While previous work showed FLIPD’s empirical success and proved its correctness only under the restrictive assumption of affine submanifolds, this work extends the analysis to more realistic nonlinear manifold settings. The authors rigorously establish the validity of FLIPD in these general cases and further show that similar results hold when Gaussian convolutions are replaced with uniform ones. These results bridge the gap between theory and practice, enhancing the trustworthiness of LID-based methods in applications like outlier and adversarial detection. \n\nBased on the following comments, the reviewer suggests a revision for this paper.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "There are some concerns and questions about the technical details in this paper. \n\n**1.**  Eq. (2) about the probability density function (pdf) of an isotropic Gaussian is confusing. Here, $x \\in \\mathbb{R}^D$ and mean $\\mu \\in \\mathbb{R}^d$ are vectors, while the covariance $\\delta$ in $\\mathcal{N}_D(x; \\mu, \\delta)$ is a scalar. Moreover, in Eq. (2), where do the terms $\\exp(-2\\delta)$ and $\\exp(-D\\delta)$ come from? \n\n**2.** The major contribution of this paper is a rigorous theoretical justification that Equation (1) remains valid when the data distribution $p$ is supported on a general disjoint union of smooth submanifolds. This result significantly generalizes previous work, which was limited to affine submanifolds, and formally supports the use of FLIPD for estimating local intrinsic dimension (LID) in more realistic geometric settings (see Theorems 1 and 2). However, a key limitation is that the paper does not provide guidance on how the derived theoretical results can be practically applied to compute LID in real-world diffusion models. In particular, it would be valuable to include discussion or examples showing how these results translate into usable estimators in practice, especially in the context of high-dimensional generative models.\n\n**3.** Building on the concern raised above, it would greatly strengthen the paper if the authors included empirical experiments demonstrating how the derived equation can be leveraged to compute the local intrinsic dimension (LID) in practice. \n\nIt is believed that the paper would be accepted if the authors could fix the above points.", "broader_impact_concerns": "No ethical concerns. The paper focuses on the theoretical analysis of existing LID estimators in diffusion models, with no direct implications for sensitive applications. The Broader Impact Statement, if present, adequately addresses the scope and limitations of the work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "QYqtye9Q71", "reviewer": "Reviewer_LhHr", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n1. The paper tackles a clear and well-motivated gap in the literature: while FLIPD (Kamkari et al., 2024b) performs well empirically, its theoretical foundations were previously limited to affine manifolds. The authors successfully extend this to general submanifolds and disjoint unions thereof, under mild assumptions.\n\n2. The uniform convolution result is a nice result. While not directly useful for DMs, it strengthens the intuition behind older ball-based LID estimators, and could be of interest to the broader theory community.\n\n## Weaknesses\n\n1. The results assume access to the exact score function; in practice, this is always learned approximately via a neural network. While the authors mention this limitation, the paper would be considerably stronger with at least a first-order bound quantifying the bias introduced by imperfect score learning (e.g., some bound on the deviation of FLIPD from LID as a function of $||\\hat{s}-s||$).\n\n2. The paper is purely theoretical and includes no empirical results. Even a small illustrative experiment (e.g., convergence behavior of FLIPD as $\\delta \\rightarrow -\\infty$ on a toy manifold) would help ground the theory in practice and potentially shed light on rate of convergence or finite $\\delta$ effects.\n\n## Verdict\n\nThis paper makes a meaningful theoretical contribution to the literature on LID estimation and generative models, in particular by completing the theoretical justification for the FLIPD estimator under realistic assumptions. While the results are formal in nature and the scope is limited to idealized score access, I believe the work is valuable and the claims made in the paper are well-supported. Moreover, I believe this paper is relevant to at least some portion of TMLR's audience, especially those working on diffusion models, manifold learning, and geometric statistics. As a consequence, I recommend an accept rating for this paper pending the completion of the few minor revisions requested below.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper provides a rigorous theoretical foundation for the Fast Local Intrinsic Dimension (FLIPD) estimator of local intrinsic dimension (LID), a method introduced in prior work (Kamkari et al., 2024b) for estimating the dimensionality of the submanifold a given data point $x$ belongs to using diffusion models (DMs). Specifically, the authors prove that the key identity underlying FLIPD, which connects LID to the rate of change of the log density under convolution with Gaussian noise, holds under general and realistic manifold assumptions, as opposed to the highly restricted affine case previously assumed in Kamkari et al. (2024b). Additionally, they show an analogous result for uniform convolutions, which provides formal justification for older ball-probability-based estimators as well. The paper is entirely theoretical and contributes to closing the gap between theory and practice for LID estimation via diffusion models.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "I have highlighted my requested changes below and have specified for each whether they are critical for securing my recommendation for acceptance:\n\n(1) Please include a short discussion or bound on how FLIPD deviates from LID when the learned score function $\\hat{s}$ differs from the true score $s$. Even a heuristic discussion or an inequality involving $||\\hat{s}-s||$ would help clarify the practical limitations of FLIPD (important but not critical for securing acceptance).\n\n(2) A toy empirical example (e.g., estimating LID on a 2D manifold in $\\mathbb{R}^3$) would go a long way toward illustrating the main ideas and convergence properties (not critical for securing acceptance, but would improve the paper).\n\n(3) I have highlighted some minor typos and corrections below:\n\n- Section 2, second paragraph: \"is given by (Särkkä & Solin, 2019)\" -> please use an in-text citation here, \\citet (instead of \\citep)", "broader_impact_concerns": "The work is largely theoretical and focuses on LID estimation via generative models. I do not think a detailed broader impact discussion is needed.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)