Why We Need New Benchmarks for Local Intrinsic Dimension Estimation
Authors: Piotr Tempczyk
Dominik Filipiak
Łukasz Garncarek
Ksawery Smoczyński
Adam Kurpisz
Venue: ICLR 2026 Poster (2026)
Keywords: Local intrinsic dimension estimation
LIDL
FLIPD
Diffusion Models
Benhamark
Normalizing Flows
ESS
Normal Bundle
NB
LID
Primary area: datasets and benchmarks
TL;DR: We show that LID estimation community needs new benchmarks for intrinsic dimension estimation and come to interesting conclusions on the performance of existing algorithms.

Abstract: Neural Local Intrinsic Dimension (LID) estimators are typically bound to domain-specific architectures whose inductive biases can yield inconsistent estimates for the same underlying manifold. Existing evaluations either use overly simple synthetic data (with known LID) or real datasets (with unknown LID), obscuring true performance. We introduce a principled benchmarking framework that (i) maps the same manifold into multiple domain representations while preserving its structure, enabling like-for-like cross-architecture tests; (ii) designs harder variants of popular datasets that target key manifold properties; and (iii) applies controlled transformations with known LID shifts to stress-test methods even when absolute LID is unknown. Across this suite, including non-trivial synthetic datasets, we show that accuracy on simple manifolds does not transfer across domains and that state-of-the-art methods fail under targeted stressors, revealing clear failure modes and areas for improvement. Data and code are available: https://github.com/DominikFilipiak/LID-Benchmarks.

Published as a conference paper at ICLR 2026
WHY WE NEED NEW BENCHMARKS FOR LOCAL INTRINSIC DIMENSION ESTIMATION
Piotr Tempezyk*
Dominik Filipiak!
IDEAS Research Institute
Adam Mickiewicz University, Poznañ University of Innsbruck Perelyn
Institute of Informatics, University of Warsaw NASK National Research Institute
Lukasz Garncarek Brainly
Ksawery Smoczyáski Synerise
Adam Kurpisz"
BFH Bern Business School ETH Zurich
ABSTRACT
Neural Local Intrinsic Dimension (LID) estimators are typically bound to domain- specific architectures whose inductive biases can yield inconsistent estimates for the same underlying manifold. Existing evaluations either use overly simple synthetic data (with known LID) or real datasets (with unknown LID), obscuring true performance. We introduce principled benchmarking framework that (i) maps the same manifold into multiple domain representations while preserving its structure, enabling like-for- like eross wesss-acchttettrre tests; (ii) designs harder variants popular datasets that target key manifold properties; and (iii) applies controlled transformations with known LID shifts stress-test methods even when absolute LID is unknown. Across this suite, including non-trivial synthetic datasets, show that accuracy simple manifolds does transfer across domains and that state-of-the-an methods fail under targeted stressors, revealing clear failure modes and areas for improvement. Data and code available: https: http//githu.... com www.:/wwww......../..........a.e www.oooiiinnililiimmm.mm Ben.hhDaaeBnareeeeeeoo
INTRODUCTION
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Table Summary of our experiments. Columns: LID-estimation aspects tested by our benchmarks; rows: neural-basel algorithms (ESS classical benchmark). Performance of methods classified following the legend - H: high, M: moderate, L: low, O: out-of-range (unassessable) (more can be found in Sec. B). Gray color marks cells where similar aspects in the method's original paper were investigated; parentheses give that paper's reported performance (assumed H if absent). We show that benchmarks; many
algorithms that passed original simple tests for many of aspects did worse aspects especially on real-world datasets (RWD) remain untested and some only partly explored.
"Equal contribution.
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Recent advances in neural methods have accelerated LID estimation. A prominent line leverages generative models Tempezyk et al. (2022 2025); Kamkari et al. (2024): Stanczuk et al. (2024); Yeats
ct al. )023); Horvat and Phister (2022) 2024 Despite the rapid development of LID estimation
algorithms, scant attention has been paid to evaluating their performance. Most existing benchmarks for assessing L.D estimation methods follow common strategies. The first involves datasets sampled from well-defined and simplistic distributions where the LID is known and well-understood. The second strategy evaluates LID estimation methods on domain datasets with unknown LID.
The main advantage of the first method lies in the well-understood ground truth of LID making the evaluation of LID estimation methods reliable. Unfortunately, those datasets fail to capture the complexity of the real-world manifolds. Moreover, most existing works do not consider edge cases such as varying manifold thickness or neighboring manifolds, nor do they analyze results way that reveals inefficiencies of their algorithms even on simple datasets like Gaussian distributions.
On the other hand, the second approach for testing LID estimation methods, which involves using real-world domain datasets, matches the desired level of complexity but suffers from a significant drawback: in most cases, the ground truth of LID is unknown. This makes impossible to assess a method's performance reliably, leading to possibly erroneous conclusions.
Our contribution Following the discussion on existing benchmarks and their limitations, we present a list of contributions to the field of testing LID estimation methods, together with the limitations that we address.
We introduce a toolbox that maps datasets into different domain representations (e.g.. images) without altering the underlying manifold, enabling controlled like-for-like tests of the same dataset across neural architectures and domains; using this toolbox, show that the common procedure of validating on simple synthetic manifolds and silently assuming similar performance across domain networks is false.
We introduce variants of the datasets already studied in the literature that pose much higher difficulty for modern neural-based methods (see Ta. While many the presented methods have been reported to achieve high accuracy, show that a more careful design datasets targeting key manifold characteristics poses significant challenges for LID estimation methods.
We bridge the gap between benchmarks based on well-understood analytical distributions and real-world datasets by employing several data transformations. As result, we able to stress-test algorithms on datasets with unknown LID by evaluating their performance before and after transformation, and by comparing it to the ground-truit -ouun LID difference imposed by the transformations. We also show how drastically sample size for real-world datasets affects tested algorithm estimates.
We show the significance of testing the algorithms non-trivial synthetic datasets, that pose much harder challenge than other types described before.
Finally, the majority of modern methods are tested different datasets, which hinders understanding of the limitations that should fuel further development. This paper provides a broader collection of datasets, allowing more meaningful analysis and comparison across various methods (see Table
In the fast-moving field of LID estimation, we believe that the results presented in this paper shed new light the construction of testing benchmarks and can amplify the impact of this work by informing and supporting the many new techniques currently being explored in research labs around the world. In particular, authors of existing methods, especially those analyzed in our study, may find this paper helpful in refining or extending their own approaches. Finally, this work also highlights the need for further development of benchmarks to keep pace with the rapid and diverse advancements in LID estimation techniques. We overview LID applications, evaluated algorithms, and related work in Sec. C [ Practical guidance on LID estimation algorithms can be found in Sec.
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2 METHODS FOR CREATING DOMAIN DATASETS
We use set of transformations and methods that can be used to create challenging benchmarks. While they in principle can work for any continuous domain (such as audio, video, EEG, etc.), we focus on the images domain in this work.
Inverse Domain Representation (IDR) The goal of this method is to bridge the gap between datasets sampled from analytical distributions with known ground truth of LID and real-world datasets on arbitrary domains. While producing an artificial dataset sampled from an arbitrary manifold is not challenging (one can embed the manifold in RDD and sample from the embedding's 's range), this becomes more complex when one wants the dataset to resemble some real-world data. We need method to embed an arbitrary manifold into the ambient space of given dataset X E in manner such that the image is "close to" X. For a formal definition of IDR and detailed discussion, we refer the reader to Appendix E were used as basis for IDR transformation, samples for such dataset are presented in the Figure
For the purpose of experiments class images from FMNIST
Monotonic Embedding (ME) The goal of this method is to assess the robustmess of LID estimation algorithms to continuously differentiable geometric deformations of the data manifold. It is partic- ularly useful for datasets with unknown intrinsic dimensionality. The procedure involves applying continuously diffcrentiable, monotonic transformations to the coordinates of the data in the ambient space, effectively stretching or compressing the geometry in a controlled manner. Notably, different functions may be applied independently to each coordinate, allowing for highly flexible distortions. As long as the derivative of the applied function remains controlled, we expect the LID estimates to remain stable before and after the transformation. This follows from general theory-ir it is fundamental result that diffeomorphisms preserve manifold dimension, and the mappings we use are diffeomorphisms.
In the other methods presented, we can similarly assert that what we actually do is applying a sufficiently regular mapping to our manifold.
Ambient Space Extension (ASE) This method alters the ambient dimensionality of a dataset without modifying the LID. Similarly to the previous approach, it is suitable for datasets with unknown LID and can be used to confirm algorithm stability on a dataset. The extension is performed by introducing new dimensions as deterministic, continuously differentiable, and monotonic functions of the original coordinates. In the case of image data, this could be achieved through deterministic upscaling techniques. More formally, in ASE we re-embed M using a map of the form x + (r. (x)) for E RRd which has a smooth inverse, namely the projection onto the first d coordinates, and thus preserves the dimension. At first glance, such modifications may appear trivial. However, when viewed through the lens of deep learning models, especially convolutional neural networks, they can introduce significant complexity. The hierarchical structure of learned convolutional features may differ considerably between models trained original versus extended datasets.
Auxiliary Dimension Injection (ADI) This method increases the ambient dimensionality of datasets with unknown LID by adding informative features derived from parametric transformations of the original data. Parameters are sampled from known distributions, ensuring the result remains structurally related to the source dataset. Notably, we apply maps that have non-degenerate Jacobians, so they are immersions and preserve the local dimensions of their domains. The increase of dimension determined by the dimension of the co-domain of the new embedding we define. The approach is highly flexible. For example, an audio signal may be filtered with random low-pass cutoff and blended with the original at a random ratio, yielding two additional, non-trin trivial dimensions. In the image domain, concatenating random pairs of MNIST digits produces samples whose dimensionality equals the sum of the individual image dimensions, as the new dataset effectively forms a Cartesian product of the original space with itself.
Manifold Synthesis (MS) We generate datasets with known intrinsic dimensionality yet complex, non-trivial geometry by applying deterministic, continuous transformations to a parameterized manifold that preserve topological and differential properties. For example for images, one can parametrize object attributes (e.g., position. orientation) and render; for audio, one can combine sample fragments via controlled parameters (start time, filter cutoff, duration, volume). The result is data with well-defined LID but with substantially more complex appearance and structure.
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3 ALGORITHM ANALYSIS DEMONSTRATED USING IMAGE DATASETS
this section, we use methods presented in related work Sec. to create datasets designed to test various interesting aspects of LID estimation algorithms, along with discussion of their construction and characteristics. The details of experimental setting can be found in Sec. Morcover, for the introduced datasets, present the estimated LID for the tested algorithms. We focus presenting one plot of interesting results for selected (often good-performing algorithms, while the remaining ones shown in Sec. B
All the results are summarized in tables in Sec.
3.1 NON-UNIFORM DENSITIES
As experiments Tempezyk et al. (2022); Stanczuk et al. 2024)); Kamkari et al. (2024) and theoretical considerations Tempozyk et al. (025) show, LID estimation for non-uniform densities may close to correct value when averaged on the whole dataset but biased in certain areas, e.g., in LIDL this bias a function of the Laplacian of the density.
For example, in LIDL, the core estimate (Theorem 2.1) expresses the logarithm of the diffused density as sum of linear term (whose slope allows recover the dimension) and an error term (the effect of which diminishes as 8 + O, but affects the final estimation as in practice we probe with concrete values 8 > 0). The magnitude of the error term can traced through the proof of the core estimate to arise from non-constancy of density and curvature of the manifold. Therefore, it is necessary to more thoroughly examine the behavior under non-uniform densities.
Gaussians (IDR) The dataset is mixture of four 5-dimensional Gaussian distributions with means located at (-3,3, O, 0), (3, -3, 0,0, (-3, -3, 0, 0, 0). (3,3, 3, 0, 0) with standard deviations 1/27, 1/9, 1/3, and respectively, transformed with the IDR method. For each point, calculated the distance from the closest mean and divided this distance by the appropriate standard deviation. Results for each algorithm are presented in Figure and 14 We would expect that matter from which component of the mixture we sample and no matter how far we are from the mixture component mode, the estimate should be equal 5. Only ESS achieved satisfying results this task.
3.2 MANIFOLD CURVATURE
Building on the motivation from the previous section and the fact that the majority of algorithms tailored for flat, uniform manifolds, aim to the behavior of LID estimation algorithms on curved manifolds. Curved manifolds are commonly present in real-world scenarios.
Spheres (IDR) We used 4 disjoint spheres S with origins located in (-3,3, 0, (3,03,0,0,00 (-3. -3 0, 0, 0), (3, 3, O, O, 0) and with radia of 1/27. 1/9, 1/3. 1 respectively, transformed with the IDR method. For each point the dataset, we calculated the distance from the four sphere origins and assigned those points to the closest sphere. Estimate distributions for each algorithm and each sphere separately are presented in Figure and The only algorithm unaffected by the curvature of this dataset was ESS.
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Figure 2: Sample from Arrows (MS) dataset.
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Figure LID a function of edge dimensions for ESS on 20-dimensional Uniform (IDR) dis- tribution between -3 and 3.
Spaghetti (IDR) The dataset used in this analysis is the spaghetti line dataset introduced by Stanezuk et al. (2024) but transformed into an image domain using IDR. It is I-dimensional manifold homeomorphic with the circle twisted and folded that it occupies k 20 dimensions. Points from this manifold are sampled as follows: 0 ~ 24(0, 2T); ) 2; sin + 1)0), for ......k k. Results are presented in Table 2 and reveal that this dataset can pose serious challenge for some algorithms. Notably, shown by Stanczuk et al. (2024), the NB algorithm could solve the dataset without IDR domain transformation up to an embedding into k = 100 dimensions with high accuracy. However, the IDR transformation introduces significant challenges, leading higher (but still reasonable) errors even for k = 20.
3.3 BOUNDARIES OF MANIFOLDS
An extreme of non-uniform density is distribution with sharp edges, like uniform distribution on a hypercube. Such distributions are common in real-world datasets due to measurement limitations. For example, cameras can't record light intensity beyond threshold, image datasets often lie within a hypercube, with many points on its boundary (any images with some white or black pixels).
Many algorithms work under the assumption that when measuring LID point x, a sufficiently small neighborhood of x where density has some nice properties, e.g., being sufficiently smooth. However, in practice, the neighborhood under consideration contained in ball of some radius r, whose center can lie the proximity of a boundary or precisely on it. This leads to a problematic case, which was observed empirically by Tempozyk et al. (2022). and formalized later by Tempezyk et al. (2025).
considering
Uniform (IDR) The dataset is 20-dimensional uniform distribution between -3 and on each dimension, transformed with the IDR method. In this test, we are grouping points that are in the proximity of m edges. A point is said to close an edge if it lies the original manifold closer than 0.25 from the edge located at 3 or -3. We test LID estimation for various values of parameter m. Results are presented in Figure [6] In the first figure, monotonic relationship between the average LID and number of edge dimensions for the ESS algorithm can be observed. The pattern is less visible for other algorithms though.
3.4 THIN MANIFOLDS
An interesting scenario arises when moving along the manifold certain path, where the local intrinsic dimension remains unchanged. However, an orthogonal direction to this movement, the manifold becomes thinner. In such cases, would expect to observe consistent LID estimations across all observations in a reasonable range. Nevertheless, some algorithms might mistakenly identify the manifold as having a lower local dimensionality.
Moon (IDR) The manifold we study is 3 dimensional. It is moon-shaped in the first two dimensions and uniform interval in the third one, parameterized by More formally, it sampled uniformly
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Figure 7: LID estimates for the Funnel (IDR)
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dataset using ESS.
random from the points (I1, T2, rs) E R intersected with the set M defined as
M = (1l( #2)] S v, (21, 0.1)]| 0.899r, [aaa| S =),
where r is a radius hyperparameter that can be chosen arbitrarily. For our experiments, we used the value 3. The resulting manifold has thickness of 0.201r the bottom part and 0.001r on the upper part. Finally, the dataset is transformed using the IDR method.
Figure 6 and show results for different algorithms. ESS has the best performance among them. It was able to maintain the correct estimate until the manifold was very thin; on the other hand, the NB estimates were close to ground truth but with errors distributed quite independently of manifold thickness. What is interesting is that we can observe a slight drop in the estimate for ESS close the border of the moon, which the same effect that was observed on the edge of uniform distribution.
3.5 NEARBY MANIFOLDS
The parallel ID manifolds example in Tempezyk et al. (2022) 2025) suggests, that when the expected euclidean distance in the ambient space to the nearest neighbor from other region of the manifold (or other manifold) is similar smaller to the expected distance to the local neighbor the same region of the manifold, it may lead to a bias in the estimate. The ability to recognize separate manifolds close each other for finite sample size is desirable property for LID estimation algorithms. To showcase this phenomenon, we introduce the Funnel and Spiral datasets.
Funnel (IDR) We consider 2-dimensional funnel embedded dimensions, visualized Figure The first two coordinates of the manifold original space: æ and z2, and generated using this set of equations and transformed with the IDR method:
t = 4100 8);
3 exp (-t); 0 ((0.2mm)
41 = 4; x2 = rsin 0; x3 = cos 0.
Results presented in Figure and 18 We observe that when the algorithm gives proper estimate when the radius of the funnel is high, for a low radius, the estimate distorted. The ESS algorithm behaves as expected. For wide parts of the funnel, the estimate exact. For the narrower parts, it goes up because manifolds are close to each other, and the algorithms start to detect points in all 3 dimensions. On the right end, goes down when it starts resemble 1-dimensional line. The rest of the algorithms give more or less biased and noisy estimates compared to ESS.
and
Spiral (IDR) data set we consider is a spiral dataset visualized in Figure X for the first two coordinates and Z2, generated using the set of equations:
(11,,111,1 100); 1/4 m rsin(1/r); x2 cos(11r))
where the distance to the closest point from the next revolution of the spiral gets smaller when go down the spiral. The dataset has usefuu property: the expected distance to the nearest neighbor calculated the manifold is the same at any point the manifold. Such constructed dataset is then transformed with the IDR method.
An interesting observation for ESS algorithm, showcased in Figure is that the algorithm shows an LID estimate close to 2 only during the first revolution, where the manifold visually has an LID equal for few more revolutions, especially when looking full dataset which is 100x bigger

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Figure 8: LID estimates for the Spiral (IDR) dataset using ESS. The test set shown in the fig- ure is 100x less dense than the training set.
Figure 9; LID estimate in sample size FM- NIST. The x-axis is shown in logarithmic scale to highlight differences in the small-sampi regime (n < 100),
than the test set. We want to highlight the fact that the algorithm was with hyperparameter of 100 neighboring points, which is standard value for this parameter. We observed our experiments that reducing the parameter responsible for the number of neighbors in ESS can reduce the estimate етrоr by giving correct estimates further down the spiral. However, this dataset dependent hyperparamettrization is a problematic approach for datasets with unknown LID that cannot be inspected visually. We also note that the ESS algorithm performed very well compared to the remaining methods, for which the results are presented in Figure 19
3.6 LACK OF NETWORK ARCHITECTURE IN INVARIANCE
In our experiments observed, that for LIDL and FLIPD use of convolutional networks in the algorithm (Glow<Knngn) and Dhariwal 2018) and U-net Ronneebrer et al.] (015)) respectively) yields worse results than using feedforward based networks (MAF(Papamakarró) MLP) on the same manifold but transformed using IDR. We can see in Table 4 that some of the effect may be just from widening the ambient space, but similar effect (2024). Our experiments for NB show that on Gaussian and Sphagetti datasets transformed by IDR we obtain different results than in the original paper without IDR transformation (more in Sec 3.2
al. 2017) and
reported in Kamkari et al.
3.7 ESTIMATED LID SAMPLE SIZE
Tempezyk et al. (2022) showed that for numerous algorithms the bias of the estimate is dependent on sample size. While it natural that an algorithm's error gets smaller for bigger sample sizes, an introduced bias leads to situation where we don't 't know if we have enough data for our estimate to be correct. To test that we created series of training datasets by drawing samples of different sizes from the FMNIST dataset. The validation set used for early stopping and the test set was held the same. One may wonder why we chose a real-world dataset rather than an artificial one with known LID. Figure in (Tempzyyk et al. (2022) shows that such dependency does not occur for LIDL on artificial data, so we aimed for more challenging dataset as our experiments demonstrated that LIDL exhibited a noticeable bias for small sample sizes the FMNIST dataset.
The results are presented in Figure 9 and 20 We observe a rather significant dependence of the estimate the sample size. Among all the algorithms, the NB algorithm achieves interesting and desirable characteristics. It stabilizes average estimate values for datasets bigger than 1000 samples, which is good result compared other algorithms. One interesting algorithm in this context Erba et al. 2019), which is designed to deal with undersampled regions, but we did not test it in our work.
3.8 REAL-WORLD DATASET TRANSFORMATIONS
For most real-world datasets used to evaluate the performance LID estimation methods, the ground truth of LID is unknown. The absence of ground truth makes it impossible to reliably assess the quality of LID estimates produced by these algorithms in such domains. In what follows, we propose of transformations a rigorous evaluation of algorithm performance by measuring the difference in LID before and after applying the transformation. In all experiments in this subsection, we used a downscaled version of
such datasets that modify dimensionality in controlled manner, enabling
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Figure 11: LID before upscaling vs LID after upscaling of the FMNIST images for ESS algo- rithm.
FMNIST as our base dataset. The original images were resized to 16x 16 pixels. Results from this section presented in Table
Added dimensions (ADI) In this dataset, we added an 8-pixel-wide frame using mirror padding in eight directions, creating images of size 32x32 32 pixels. We generated three variants of the dataset with 0, 4, or 8 added dimensions by applying random brightness changes to respective number of reflections. The results of LID estimation for the ESS algorithm are presented in Figure [O] The estimates for datasets with added dimensions were calibrated by subtracting the number of added dimensions from the respective estimates, ensuring that the results should align with the identity line.
For ESS, points where LID) estimation indicated lower dimensionality in the original dataset remain close to the identity line, suggesting a small relative error after image transformation. In contrast, for points where LID estimation the original dataset high, the estimates after transformation fall significantly below the identity line, indicating large estimation errors. The LIDL algorithm, in turn, adds dimensions even when the dataset has zero additional noisy reflections, meaning no additional dimensions were introduced. NB maintains the estimate on the modified dataset close to the identity line for some points, but for others. it overestimates the LID regardless of the number of added noisy reflections. Another observation is that the estimates vary significantly, sometimes by more than +550% compared the estimate the base dataset. All results presented in Figure 21
Upscaled (ASE) The dataset used was an upscaled 32x32 version of the base dataset. For upscaling, we used a torch function interpolate with bilinear mode. NB's performance in this task is quite impressive, despite the variability of the estimate, which remains of a similar magnitude as in the previous experiment involving added dimensions. ESS returns estimates lower than the original ones, while FLIPD and LIDL exhibit the opposite behavior, outputting higher estimates. All results are presented in Figure and 22
Stretched (ME) We create ME by using polynomial transformation applied to each pixel after normalizing its values to the range (0, 1). We performed two transformations using different exponents: [0, 1] + y = x², for E {0.25, 4). The best west-performing algorithm in this case is ESS, which maintains a similar mean LID before and after the transformation, albeit with high variance. NB and LIDL estimates drop significantly after the spatial transformation, while FLIPD heavily overestimates the LID after transformation. Full results in Figure 12 and
3.9 REAL-LIKE DATASET WITH KNOWN LID
There are almost datasets that simultaneously resemble real-world images and have known underlying LID. Two notable exceptions are the Gaussian blobs from Stanczuk et al. (2024) and 3DIdent dataset introduced by Zimmermann et al. (2021). The latter computational budget and available GPU resources. Therefore, we created similar dataset using MS approach.
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Figure 13: LID distribution estimated using NB for different manifolds in arrows dataset. Mani- fold dimensionality in legend.
Arrows (MS) The dataset consists of 32x32 images of arrows placed black background. Each arrow is described by six variables: horizontal and vertical position, rotation, and color (three variables for RGB). The manifold dimensionality is six times the number of arrows in the image. A sample is shown in Figure 2 There is small possibility of manifold collapse when arrows perfectly overlap, but the probability of this occurring is less than 10-3, making it negligible.
In this experiment, none of the algorithms produced results close to the ground truth. The ESS estimate failed to distinguish between different manifolds, consistently outputting values around 15 for all cases, with variations appearing only in an uncontrolled manner. The NB algorithm significantly overestimated LID values and produced some outlier estimates for LID of value around 3k. FLIPD produced estimates ranging from -20 to 40. Figures 13 and 24 present full results. We comment this results in Sec.
ANALYSIS OF ALGORIHTMS PERFORMANCE
ESS The ESS algorithm performed very well for datasets with dimmensiiionl manifolds. From experiments of [Tempzykk et al. (022) we know, that its performance deteriorates if manifold dimensionality raises, which can observed on 20D uniform datasets. We observed preliminary experiments that its behavior can change when working on smaller samples, and the n_nei ghbours parameter is crucial to the performance, but there no prior way setting it right. We could go with the highest value possible due to the computational and memory constraints, but in the case of the Spiral dataset the smaller values works better, especially for smaller sample sizes like 10K, so there no right answer that problem.
NB Performed well, especially compared to other algorithms using neural networks. It failed on arrows, stretch, gaussian and spheres and had higher error than ESS low-dimensional manifolds. will surely beat ESS on higher-ding higher-dimensional manifolds due to the ESS underestimation bias for higher-dimensions, but compared ESS it lacks the precision and robustness on simpler manifolds. This perhaps may be corrected by the person skilled in training diffusion models, but in practice this method will benefit from more stable and less noisy estimates, especially because some tests were failed due to the high variance and not high bias like in the case of other methods like LIDL and FLIPD.
LIDL This algorithm performed much worse than expected based on its performance on datasets with known LID from the original paper. the first experiments, we used LIDI with Glow Normalizing Flow Flow Kingma and Dhariwal (2018), but it performed worse than MAFiPapamakarios et al.) (2017). and its training time was an order magnitude slower than MAF so finally we sticked with MAF, To further investigate this algorithm, before the IDR transformation, before the IDR transformation but padded with zeros to reach 784 ambient space dimensions, and after the IDR transformation. This analysis provided deeper insight into where the accuracy of LIDL's estimates deteriorates. In the first case, where the original ambient space before IDR was 30-dimensional, LIDI. performed well. However, the IDR transformation
ran LIDL on selected datasets in three different scenarios:

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expanding the ambient space caused LIDL's performance to deteriorate. The results of these experiments are presented in Table 4
Yet there is another problem with LIDL: although thcoretical results in Tempozyk et al. (2025) show that when 8 - we should get an unbiased estimate, in practice we always get the estimate close to ambient space dimensionality. This is an unsolved problem of how to choose this parameter in real-world scenarios. Those results are presented in Sec. H. where we can observe how much the LIDL estimate varies with ö. For presenting the results we have chosen one range which had the best performance on IDR datasets, still not being even close to the underlying LID value.
FLIPD This algorithm suffered from the same problems LIDI,, but greater extent. Authors of the FLIPD describe a "knee" on the plot where the estimate is the closest to the ground truth, but in practice, those models many times had problems converging and producing unreliable estimates with a hard-to-find "knee" structure. When we knew the underlying LID we were able to present better results when choosing the value of with smaller MAE, but it is not a practical scenario for real-world datasets. The results for different values of t are presented in Sec. H where we can observe how much FLIPD estimate varies with L.
Authors of Tempezyk et al. (025)) show that the theoretical foundations behind LIDI and FLIPD are solid and give tools to calculate reasonable ranges of delta for the given problem, so it suggests that the problem is the non-ideal density estimator.
5 CONCLUSIONS AND FUTURE WORK
This paper shows that there are many aspects of LID estimation that should be investigated while working on new algorithms in the future. The results strongly suggest that the fomain-adapted benchmarks are a crucial aspect of this process: very similar algorithms in terms of performance on the classical benchmarks turned out to be very different when tested in new way. There are many aspects that are not covered in this work and should be further investigated, like the effects of quantization of the dataset on LID estimate, testing on higher-dimensione with the noise in the data. There is also a need to test similar algorithms in other domains, such as audio and other real-world datasets similar the arrows of 3DIdent. We hope that our work will be an inspiration for future research and will aid progress in the area of LID estimation.
manifolds, and dealing
Although we provide detailed, plot-based evaluations of each aspect and method, methods improve there will be a need to quantify each aspect with some metric in a principled way. But we believe that are not there yet and current algorithms still require refinement to the point which majority of scores reach at least M in [able
Acknowledgements. This work was partially supported by Sonata Bis grant 2024/54/E/SS6/003888 awarded by NCN Poland.
CRediT Author Statement. Piotr Tempezyk (29% of the work): Conceptualization, Methodology, Software, Validation, Formal analysis. Investigation, Data Curation, Writing - Original Draft, Writing Review & Editing, Visualization, Supervision, Project administration. Dominik Filipiak (28% of the work): Investigation, Software, Data Curation, Writing Review & Editing, Visualization. Lukasz Garncarek (9% of the work): Conceptualization, Methodology, Formal analysis, Investi- gation, Writing Original Draft, Writing Review & Editing. Ksawery Smoczyúski (5% of the work): Software, Data Curation, Writing Review & Editing. Adam Kurpisz (29% of the work): Conceptualization, Methodology, Validation, Formal analysis, Investigation, Writing Original Draft, Writing - Review & Editing, Visualization, Supervision.
Reproducibility Statement. We provide the dataset along with source code and scripts required to generate it on GitHub: htt : /tttps:://itti /github. com/ As for LID algorithms, implementation details and hyperparameters are described in Appendix F
/LIII LIDD- Benchmarks
Ethics Statement. This work concerns methodology for Local Intrinsic Dimension (LID) estimation, Topological aata-analysis problem. It does not involve human subjects, personally identifiable information, or sensitive attributes, and it introduces no foresecable direct societal harms. All datasets used are public or synthetic, and we follow their licenses and terms of use.
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Francesco Camastra and Antonino Staiano. Intrinsic dimension estimation: Advances and open problems. Information Sciences, 328:26-41, 2016.
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APPENDIX
A TABLES
Table 2: LID estimations with MAE for the datasets with known dimensionality.
algorithm
ESS
FLIPD
LIDL
NB
MAE
MAE
MAE
MAE
Gaussians
4.99
0.09
0.07
3.25
3.57
3.08
17.24
3.09
12.24
5.72
0.66
0.81
Spheres
5.07
0.08
0.09
4.49
3.77
3.24
17.90
3.77
12.90
6.80
1.06
1.83
Spughetii
Uniform
18.87
0.43
1.13
18.08
3.30
2.80
34.70
3.61
14.70
19.41
1.78
1.07
Moon
2.86
2.14
12.31
9.31
2.75
Funnel
2.20
0.26
0.21
1.73
3.46
3.24
14.18
3.09
12.20
1.48
0.51
0.54
Spiral
1.99
0.13
0.99
1.47
3,36
3,34
1.98
5.38
2.26
2.03
0.73
1,03
Arrows
14.73
3.88
6.23
8.57
25.80
17.43
455.85
1,007.53
440.82
Table 3: LID estimations for the modified real-world datasets with unknown dimensionality.
algorithm
ESS
FLIPD
LIDL
NB
FMNIST (base)
15.32
3.75
55.92
24.62
138.73
44.11
133.45
38.34
FMNIST (add dim +Od)
14.87
3.70
210.03
45.92
227.01
62.37
142.40
37.57
FMNIST (add dim, +4d)
15.21
2.40
166.43
45.91
235.48
67.07
144.89
37.68
FMNIST (add dim, +8d)
16.72
1.65
193.58
38.85
257.39
61.89
154.77
38.91
FMNIST (upscaled)
12.86
2.67
129.57
38.12
197.13
65.71
132.78
39.44
FMNIST (stretched x 2.25)
14.92
4.44
49.96
23.21
105.17
42.25
98.10
58.18
FMNIST (stretched x²)
17.98
4.85
86,355.13
94,294.77
81.68
40.36
85.74
43.34
Table 4: LID estimations with MAE for the datasets with known dimensionality.
algorithm
LIDL (w/ IDR)
LIDL (org. manifold)
LIDL (org. manifold + padding)
MAE
MAE
MAE
Gaussians
17.24
3.09
12.24
4.34
1.17
0.90
18.68
19.38
22. 14
Spheres
17.90
3.77
12.90
4.69
1.10
0.83
27.49
12.99
23.28
Spaghetti
10.25
6.40
9.53
5.30
1.94
4.31
36.25
11.41
35.25
Uniform
34.70
3.61
14.70
19.57
1.06
0.88
32.46
10.59
13.58
Moon
12.31
5.45
9.31
2.97
0.31
0.24
-21.31
18.99
25.57
Funnel
14.18
3.09
12.20
1.87
0.61
0.54
11.20
5.98
9.56
Spiral
1.98
5.38
2.26
0.85
0.88
0.66
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B ADDITIONAL RESULTS
B.1 NON-UNIFORM DENSITIES
5.1
0.56
6.0
5.0
5.8
1.00
5.6
5.0
5.4
Mean
5.0
Moan
5.2
5.0
5.0
5.0
4.8
4.6
0.5
0.8
1.0
1.2
1.5
1.8
2.0
2.2
2.5
0.5
0.8
1.0
1.2
1.5
1.8
2.0
2.2
2.5
r/0
7/0
(a) ESS
(b) NB
0.004
5.0
20.0
4.5
19.0
1.00
Mean LTD
18.0
Moan
3.5
16.0
1.11
15.0
2.5
2.0
0.5
1.0
1.5
2.0
2.5
0.5
0.8
1.0
1.2
1.5
1.8
2.0
2.2
2.5
7/0
(c) LIDL
(d) LIDL (org. manifold)
5.0
4.0
3.0
2.0
1.0
0.5
0.8
1.0
1.2
1.5
1.8
2.0
222
2.5
1/a
(c) FLIPD
Figure 14: Results for gaussians dataset for different algorithms. ESS ranked H because estimate is almost perfect; NB ranked L because it overestimates for most of the cases; LIDL ran on original manifold yields accurate estimates for more than half of the cases, ranked it M.
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B.2 MANIFOLD CURVATURE
0.5
0.0
5.2
Iisumated LTD
Estimated
(a) ESS
(b) NB
0.2
0.1
0.0
10
15
20
25
Estimated
(c) LIDL
(d) LIDL (org. manifold)
0.25
0.00
0.2
0.0
0.1
10
Esimated
FLIPD
Figure 15: Results for spheres dataset for different algorithms. ESS ranked H because estimate is almost perfect; NB ranked L because it overestimates for all of the cases; LIDL ran on original manifold yields accurate estimates for around 30% of the cases, ranked it M.
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B.3 BOUNDARIES OF MANIFOLDS
19.0
19.6
19.0
19.6
18.9
19.5
18.9
Mean
19.4
18.8
18.8
19.4
18.7
19.3
18.7
Numbor edge dimensions
edge dimensions
(a) ESS
(b) NB
19.7
34.9
19.6
Mean LTD
34.8
Mean
19.5
34.6
19.4
34.5
19.4
34.4
19.4
Number of edge dimensions
Number of edge dimensions
(c) LIDL
(d) LIDL (org. manifold)
18.2
18.0
17.8
17.6
17.0
Number edge dimensions
(e) FLIPD
Figure 16: Results edge points from uniform dataset for different algorithms. ESS underestimates the true value in all cases, most of the points further from edges it underestimates so we ranked M: LIDL on original manifold underperformed only in cases, where points close to 4 or more edges, ranked it H.
ranked it L; NB has better estimate when closer the edge, but for
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B.4
THIN MANIFOLDS
3.0
3.0
3.0
2.75
2.8
2.50
1.0
1.0
2.25
0.0
0.0
2.00
2.4 2admmm
110
-1.0
1.75
2.2
1.50
22.0
2.0
1.00
11
2
-2
-1
(a) ESS
(b) NB
3.0
45
3.0
3.75
40
2.0
3.50
35
3.25
1.0
1.0
30
3.00
0.0
25
2.75
1.0
20
-10
2.50
2.0
15
2.0
330
3.0
2.00
-2
(c) LIDL
(d) LIDL (org. manifold)
3.0
2.0
7.5
1.0
Eetmmttd LID
2.5
0.0
1.0
-25
550
-3.0
-75
3
2
(c) FLIPD
Figure 17: Results for moon dataset for different algorithms. We ranked ESS H due to the accurate estimates for most of the time; We ranked NB and LIDL. on original manifold L, due to the high variance in the estimate.
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B.5
NEARBY MANIFOLDS
Funnel
3.00
2.8
2.0
2.50
2.6
1.0
1.0
2.25
2.4 Comman
0.0
2.00
1.75
11.0
-1.0
2.0
1.25
-3.0
3.0
1.00
-2
(a) ESS
(b) NB
25
3.0
3.00
2.75
2.0
20
2.50
Estamata aboa LID
1.0
15
2.25
Falimated
0.0
10
0.0
2.00
1.75
-1.0
1.50
2.0
1.25
330
3.0
-2
2
-4
-2
2
(c) LIDL
(d) LIDL (org. manifold)
2.0
1.0
4
Estimated LID
0.0
2
-.0
-2.0
-3.0
-2
(e) FLIPD
Figure 18: Results for the funnel dataset for different algorithms. We ranked ESS H due to the accurate estimates for most of the time; We ranked NB and LIDL on original manifold L due to high variance in the estimate.
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Spiral
2.0
3.00
0.4
1.8
2.50
0.2
Aimmmlaa LTD
0.2
1.6
0.0
0.0
2.00
-0.2
0.2
1.75
1.50
0.4
1.2
-04
1.25
-0.6
0.6
1.00
-0.25
0.00
0.25
0.50
0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
(a) ESS
(b) NB
30
25
2
Balimatted
15
10
-0.4
-0.50
-0.25
0.00
0.25
0.50
0.75
-0.50
-0.25
0.00
0.25
0.50
0.75
(c) LIDL
(d) LIDL (org. manifold)
Estimated LID
0..6
-0.50
-0.25
0.00
0.25
0.50
0.75
(c) FLIPD
Figure 19: Results for the spiral dataset for different algorithms. We ranked ESS M due to the accurate estimates at the beginning the spiral; We ranked NB and LIDL on original manifold due to the high variance in the estimate.
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B.6 SAMPLED FMNIST (LID VS SAMPLE SIZE)
20.0
800.0
19.5
700.0
Mean LID
Moan LID
18.5
600.0
18.0
500.0
17.0
16.5
400.0
10a
102
10a
10c
10
102
10a
104
Sample size
Sample size
(a) ESS
(b) NB
800.0
400.0
Mean LID
700.0
Mcan LID
300.0
600.0
500.0
100.0
101
102
10a
10a
10
10a
10a
104
Sample
Sample
(c) LIDI
(d) FLIPD
Figure 20: Per-method results for various sample sizes from FMNIST dataset. We ranked ESS L because the estimate dependence sample size looks worring; We ranked NB H because the estimate stabilizes with sample size; ranked LIDL L, because the estimate do not stabilize as sample size grows.
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B.7 ESTIMATE INVARIANCE DATASET TRANSFORMATION
B.7.1 ADDING ARTIFICIAL DIMENSIONS
Estmmled LTD (modiied datuse))
With dimonssons
Estimlted LID (modllied dalaset)
25.0
With diimeniion adied
20.0
10.0
With no dimension- ndéwc
5.0
With 4 dimensions udded
With dimensions added
100.0
Estimated LID (original datasel)
Estimated LID (original dataset)
(a) ESS
(b) NB
Estiatedd LTD (medilled dalaseU
Estimated LTD (modified dalasol)
500.0
600.0
400.0
500.0
300.0
400.0
300.0
200.0
200.0
100.0
dimeasions
100.0
With dimensions added
dimensions
With dimensoons added
dimonstions
100.0
200.0 300.0 400.0 500.0 600.0 700.0
0.0
100.0
200.0
300.0
400.0
500.0
Estimated (orginal dataset)
Estimated LID (original dataset)
(c) LIDI
(d) FLIPD
Figure 21: Estimated LID for FMNIST datasets with extra artificial dimensions. Estimates for datasets with added dimensions were calibrated in way, that number of added dimensions were subtracted from respective estimates, so that on average results should lie on identity line (y = X). We ranked NB M because it is the only algorithm that have some points at identity line.
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B.7.2 UPSCALING
Estimated LID (modiled datuset)
Upscaled
Estimated LTD (modified detase!)
Upaculed datanet
25.0
200.0
20.0
150.0
15.0
100.0
10.0
5.0
50.0
10.0
15.0
20.0
25.0
50.0
100.0
150.0
200.0
Estimated LID (original dataset)
Estimated LID (original dataset)
(a) ESS
(b) NB
Esimated LTD (modilted dalasel)
Upacaled Gatesel
Essiaated LID (moditted dataset)
Upecided dateest
300.0
200.0
300.0
150.0
200.0
100.0
50.0
100.0
100.0
200.0
300.0
400.0
500.0
0.0
50.0
100.0
150.0
200.0
250.0 300.0
Estimated (original dataset)
Estimated (original datasel)
(c) LIDL
(d) FLIPD
Figure 22:
Effect of upscaling on FMNIST data. We ranked NB because it the only algorithm that have some points at identity line.
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B.7.3 SPATIAL TRANSFORMATIONS
35.0
1.023
200.0
30.0
estimate on
25.0
estimate
20.0
100.0
15.0
10.0
50.0
5.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
0.0
50.0
100.0
150.0
200.0
LID estimate on a (original dataset)
LID estimate on (original dataset)
(a) ESS
(b) NB
300.0
1=0.2
LID estimalte on
400000.0
LTD estimale
200.0
300000.0
150.0
200000.0
50.0
100000.0
0.0
50.0 100.0 250.0 350.0
100000 300000 400000 300000000000000000
LID estimate on a (original dataset)
LID estimate on (oriiinal dataset)
(c) LIDL
(d) FLIPD
Figure 23: Estimated LID under FMNIST spatial stretching transformation. Exponent of the transformation is in the legend. We ranked ESS M, because despite the high variance, at least the points were distributed approximately in equal numbers both sides of the identity line.
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B.8
REAL-LIKE DATASETS WITH KNOWN LID
0.5
0.1
0.05
0.0
0.00
0.1
18
18
0.025
0.0
0.000
24
0.0
0.00
175
Estimated LID
Estimated LID (lmited to 200)
(a) ESS
(b) NB
0.025
0.000
0.02
0.00
24
0.02
0.00
400
-20
Estimated LID
(c) FLIPD
Figure 24: Estimated LID for arrows dataset. We ranked ESS and FLIPD as L, only because least some of the estimates are close, but we can see that with high probability it is purely by accident and because those algorithms are performing well on this task.
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C RELATED WORK
LID applications There has been surge of interest in estimating the Local Intrinsic Dimension (LID) manifolds embedded in higher-dimensional space due to its relevance various tasks,
including representation learning (Ansuini et al., 2019 Li et al., 2018 Rubenstein et al. 2018).
dimensionality reduction, and clustering (Vapnik] 2013 Kleindessner and Luxburg 2015 Camastra and Staiano 2016, Loaiza-Ganem al. (024). LID estimation is crucial for applications like
manifold learning, density estimation (Brehmer and Cranmer) [020] Caterini et al. [021] Ross and
Cresswell 202I), and generative auto-encoder models such as VAE (Kingma and Welling (2014). These methods often rely on a predefined manifold dimension as a hyperparameter, and accurate LID estimation is critical. RRubenstein et al. (2018) demonstrate that errors in estimating the latent space dimensionality can affect the performance of these methods. LID estimation was used as an analytical tool study the process of training and representation learning in deep neural networks (Li al. 2018 Ansuini et al. 2019). Results of Pope et al. (2020) reveal that the intrinsic dimension of dataset impacts the training process of a machine learning model, sample efficiency, and its ability to generalize. Ross al. (2024) show that LID estimation can be used to investigate memorization in generative models. The earliest methods for estimating manifold dimensionality primarily employed global approach, pioneered by Fukunaaa and Olsen (1971) and later developed in works such as Minka (2000); Fan et al.] (2010). Recent approaches address the problem locally by analyzing neighborhood geometry in non- non-parametric manner Johnsson al. (2014) Levina and Bickel (2004).
LID estimation methods Over the past decades, numerous methods have been proposed for LID estimation, following various methodologies. CCaassrra and Staiano (2016) examined the older methods in a comprehensive survey. In our work we test four methods for LID estimation: ESS Johnsson et al. (2014), L et al. (2024) across the datasets introduced this paper. We wanted to compare neural methods with older non- conamerrmttric one, and ESS was the best performing algorithm among classical methods like MLE (Levina and Bickel, (0044) according to experiments in [Tempzyyk et al. (2022). We briefly describe tested algorithmss in Sec. D
et al. (2022), NB Staanzzk et al. (2024), FLIPD Kamkari
D ALGORITHM DESCRIPTION
Johnsson et al. (2014) This method assumes that the data lie in a local neighborhood of larger set with relatively small curvature and noise. the ideal case, where there is no noise or curvature, this corresponds to data points being uniformly distributed in a hypersphere. ESS estimates the intrinsic dimension by analyzing the distribution of angles between data vectors. It provides low-bias estimates even when the intrinsic dimension exceeds the number of data points.
LIDI Tempozyk et al. (2022) This algorithm uses the rate of change the probability density under the Wiener process estimate the local intrinsic dimension (LID). For small diffusion times the logarithm of the density is a linear function of log t. The slope of this linear relationship is D, where d is the intrinsic dimension of the manifold and D is the dimension of the ambient space.
NB Stanczuk et al. (2024) NB is based on the observation that for small diffusion times, the score function of a diffusion model lies in the normal space of the manifold. Using this fact, authors estimate LID at given point by applying a singular value decomposition (SVD) to assess the rank of relevant matrix constructed from the score function.
FLIPD Kamkari et al. (0044) FLIPD is a more scalable variant of LIDL, using diffusion models to estimate density.
E IDR FORMALIZATION
Take, for example, dataset X C RD of face images, and manifold M C Rd, already embedded in Rd. If we take d eigenvectors of the covariance matrix of the distribution of face images computed from X, their linear combinations will yield d-dimensional space of face-like images in which can embed a copy of M. Sampling from this copy gives us a dataset of face-like images arising from preseribed manifold.
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More formally, suppose given manifold . smoothly embeddable in R through Φ: M R Given any dataset X embedded in RD with D 2 d, we may artificially create dataset diffcomorphic to M, interpolating the points of X. The procedure starts with computing the mean μх RR of X and applying PCA to the centered dataset X px- Then, we take the principal component vectors w1,..-. utd RR. corresponding to d largest eigenvalues. After that, embed M in R.D through é(p): M + I. D given by the formula k(p) μx + Ei=1 (;(p),, Finally, we take 6(M) as the new (continuous) dataset, from which we may now sample points.
The embedding à is just the composition of Φ. followed by the embedding of R into RD taking the standard basis to the vectors u, Note that the vectors Wi..... Uu are unit vectors; therefore, the geometry of the manifold remains unchanged. Nevertheless, the quality of the outcome, measured by visual similarity to the target image domain, is highest when the first d eigenvalues are large, i.e., the original dataset X is not "squeezed" in the corresponding directions.
While the technique is applicable to any target domain, in this paper, we present it for image representation only,
find it the most insightful and the closest to real datasets which LID) estimation methods have been tested the past.
In this work we use images from class of FMNIST dataset to fit PCA with D 784. A sample from such dataset is depicted in Figure More discussion on a choice of a dataset to fit PCA can be found in Sec. G
Alternative approach to HDR A possible alternative for the Inverse Domain Representation could utilize the vectors Uj..... dd scaled with the corresponding eigenvalues. This approach would ensure that the resulting dataset visually better matches the target image domain; however, it could alter the geometry of the dataset, especially when the first d eigenvalues are significantly different in magnitude. In extreme cases, this could lead to the deformation of the dataset and affect the LID of the transformed data.
For this reason, since preserving the LID of the transformed dataset is crucial for our study, we do not follow this approach. Moreover, the results presented in this paper demonstrate that a transformation that preserves the geometry of the dataset while mapping it to another domain still leads to different LID estimations by neural network- -based algorithms, posing a significant challenge for the evaluation of modern methods.
F EXPERIMENTAL DETAILS
For all datasets with known dimensionality, we used 100,000 samples for the training set, 1,000 for validation, and 1,000 for test. The only exception was the Arrows (MS) dataset, which uses 100,000/10,000/ 0000000,00000000000000 split.
ESS setup In our experiments we used ESS implementation from Bac et al. (2021) which can be found under in the text. Because this implementation cannot calculate LID on unseen data and we can only get predictions for the training set, we have made decision jointly train on training and test datasets, but only present the results for the test part of the data.
with default hyperparameters if not stated otherwise
NB setup For NB experiments, we used the PyTorch implementation along with the en- vironment setup provided by (Stanczuk et al. (2024), which is publicly available under giihub.oo/(BBAA7711///D---
The conducted experiments have been carried out using DDPM Ho et al. (2020) with Adam optimizer (a = 2e-4, 4, B = 0.9, = 1c-8). The convergence has been assessed using the validation holdout dataset. For a more detailed description of hyperparameters for all conducted experiments (except for Arrows), please refer to MNIST/config.pp which is available in the aforementioned repository. The only hyperparameter adjustments we made were connected the shape of the input data. For the Arrows experiment, we used based our config the cclebAAAddpppy file.
LIDL setup To perform experiments with LIDL, we utilised the official PyTorch implementation www.iau..ooooppiimmmmt sh/lid)) released by Tempezyk et al. (2022). All the experiments (except the ones the original manifold and padding) has been performed on MAE Papamakarios et al. (017
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with 5 layers and 5 hidden units. For the version with the original manifold and padding. we had to went down with the number of layers to 4 due to unfavourable scalability and performance constraints. Each experiment has been perofrmed with the diffent set of 8 triplets, such that å {(2- (2-(m-1),2"* 2- 2 2-(+1)) In E
The rest of the hyperparameters remained in line with the version presented by Tempezyk et al. (2022).
FLIPD setup To obtain the results presented in this article we utilized fork from August 12, 2024, of the repository listed under under|github.com cttttuu.coom/ayerrallabb///////a (2024). As for now authors recommend utilizing github.coo/layeee.e com/aayeroaillabs/dgm geometry for LID estimation experiments. For all datasets, the architecture of the diffusion model was the same up to imput layer dimensionality. We used the same MLP architecture for diffusion models as authors which is described in Section C appendix of FLIPD article Kamkari et al. (024). For all datasets we ran 1000 epochs of training for each network, choosing the best model measured by its' validation loss. For the datasets with known LID values we chose which yielded lowest MAE error given all samples from test set. For datasets with unknown LID values we used heuristic presented with original article with the caveat that searched for the knee for values of larger than the maximum average estimated LID. We did it as for many of the datasets for certain t values instabilities occurred which resulted in extreme LID estimates. Those values prevented the kneed package from performing correctly as it assumes monotonicity of the function.
by the authors of FLIPD Kamkari et al.
Table 5: Approximate duration for model training on RTX 2080 Ti GPUS.
Dataset
NB
FLIPD
LIDL
FMNIST (upscaled)
1d @ 2GPU
5.5h @ 2 GPU
9h @ 1 GPU
FMNIST (downscaled)
0.5d @ 2GPU
4h @ 2 GPU
6h @ 1 GPU
FMNIST (stretched )*)
1.5d @ 2GPU
4.5h @ 2 GPU
7.5h @ GPU
FMNIST (stretched _ )
2d @ 2GPU
4.5h @ 2 GPU
7.5h @ GPU
FMNIST (add dim, +4d)
2d @ 2GPU
5h @ 2 GPU
8h @ 1 GPU
FMNIST (add dim, +8d)
4d @ 2GPU
6h @ 2 GPU
10h @
GPU
Spiral
5.5d 2GPU
10.5h @ GPU
22h
GPU
Uniform
3.5d @ 2GPU
6h @ 2 GPU
9.5h @ GPU
Funnel
4.5d @ 2GPU
6h @ GPU
9.5h @ GPU
Moon
5d @ 2GPU
6.5h @ GPU
10h @ GPU
Gaussians
1d @ 2GPU
7.5h GPU
11h
GPU
Spheres
1d @ 2GPU
7h @ 2 GPU
10h @ GPU
Spaghetti
1 1.5d @ 2GPU
6h @ 2 GPU
9h @ 1 GPU
Arrows
10d @ 8GPU
10h @ 2 GPU
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G DISCUSSION & IMPORTANT REMARKS
During the development of the article, we received numerous helpful questions and suggestions from the scientific community, for which we grateful. As many of them led to an interesting discussion and remarks, we publish an excerpt of them in this section.
Validity of the IDR transformation. Our claims regarding the properties of IDR are supported by the theory behind principal component analysis. Since PCA is an affine transformation that only rotates the manifold and translates it within the ambient space, it does not alter the key geometric properties we consider. For example, this transformation preserves distances between points and does not affect curvature- both of which crucial factors in many of our experiments.
Why did we choose the 7th class from FMNIST for IDR? When we apply PCA to most of the image datasets, the resulting representations no longer resemble meaningful images. They often appear as chaotic blobs of light and dark, lacking coherent structure. This occurs because applying PCA to a complex manifold does not guarantee that the resulting distribution for first n vectors will be Gaussian. While PCA centers and rotates the data, the resulting distribution often contains holes. PCA is used in reverse as generative model i.e., by sampling from standard normal distribution in the reduced space and applying the inverse PCA transformation-sampe
drawn from these holes can lead to unnatural and noisy outputs. This is precisely why we chose a specific shoe class from FMNIST: it was one of the few datasets where the data was sufficiently dense in the first 20 PCA components to yield visually coherent and plausible images. In contrast to other classes, this particular FMNIST category produced sharper edges and more meaningful shapes, making it suitable for our experiments in the image domain. Another reason we did not test other classes was due to computational budget constraints. Training datasets with higher dimensionality, especially RGB images instead of grayscale, would be several times more expensive an overhead we could not accommodate within our current resources.
Why do we assume that LID is a homogenous integer? In this paper, we adopt a definition of LID that aligns with those used in recent neural-based LID estimation algorithms we analysed. While the assumption of integer-valued LID is a meaningful limitation that deserves future attention, we observe that current algorithms still struggle even with relatively simple manifolds under this assumption. Addressing these remains our current focus. That said, our IDR dataset generation approach is capable of embedding any manifold into the data space while preserving its structure, making suitable for generating data with non-integer dimensionalities manifolds with singularities when we generate one as an input.
Our work focuses specifically the LID estimation problem. This perspective may not capture all the complexities relevant to downstream applications in representation learning, some of which may benefit more from global dimensionality measures. However, there is clear and ongoing interest in local LID estimation. Our goal is to support this active line of research by offering targeted benchmarks that reveal key weaknesses in existing estimators, many of which, our results show, struggle on specific, controlled instances. While the broader issues of non-integer and heterogeneous LID values are important, addressing them comprehensively lies beyond the scope of this work. Nevertheless, we believe our contributions serve as a useful and timely step toward improving future methods. Finally, we note that although many current estimators assume local homogeneity of density, our aim is not to critique accepted algorithms but to provide tools that help the community develop more robust approaches. It is also worth mentioning that some methods, such as LIDL, are already capable of handling non-uniform densities to some extent (Tempezyk et al. 2025).
Fourier-based alternative to PCA. An anonymous reviewer suggested using Fourier or DCT bases, which is indeed very interesting. They suggested to consider 2D discrete Fourier expansion of an N x N image in a form of
N-1
x (i.j) = Σ X(u, v)e 2nd(( (1.5) E (O...., - 132
0,
and retain bases with a coefficients having a high X(, v), which can proxy the top PCA directions for smoother, reqquency-oriented chart.
We agree such approach seem to be suited to the specific case of image datasets. However, the framework suggested by the reviewer is also more complex. What we need is single d-dimensional
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orthonormal system for the whole dataset, which is the case, c.g., in the PCA approach. In the proposed approach, we need to choose a subset of a larger orthonormal basis B, given something may roughly describe "compatibility score" for each sample from the dataset and each basis vector (the coefficient in the expansion of the sample in basis B). Then there is an additional step of aggregating this information to obtain single orthonormal system. Given the expansions t Eee (I, 6)6 of every in our dataset, and a fixed E B we could look at the functions æ + (I, b) and try to use their norms, e.g. e' or Ls norms. We could also use these coefficients to perform voting on basis vectors. Nevertheless, despite discussed shortcomings, IDR was enough show underperformance of LID algorithms, which was one of our goals.
Why the Arrows dataset (MS) the most challenging one? This dataset more challenging for existing algorithms because the manifold has some nasty properties that others presented don't. E.g. our theoretical considerations show that the manifold has many V-shape corners as an artifact of translating and rotating the arrow on the image. This may pose a challenge for the existing algorithms designed with the simpler manifold shapes in mind.
What is the underlying manifold in Arrows dataset, what the parameterization looks like, and how the transformation preserves the intended ID? In the Arrows dataset, when two arrows overlap, their color vectors are added without clipping, leading to unbounded RGB values, but preventing information loss. An image of a single arrow can be parametrized by the manifold M = [0, 1 x S!, where the first five dimensions correspond to (normalized) 3 components of RGB color and 2 coordinates of the center of the arrow. The St factor corresponds to the rotation of the arrow. For arrows, the image is parametrized by M* and there mapping f: M + A. where A is the space of all k-arrow images. This mapping is not a bijection, but is close enough to bijection for LID purposes. That is, outside the coordinates are distinct, i.e. every arrow has different color, position, and angle. What we claim that the restriction of f to this set is a smooth bijection with image.
union of lower-dimensional submanifolds of M all
How does local density in the funnel manifold affect LID estimation? We acknowledge that variations in local density can indeed affect the final LID estimate, as demonstrated in our experiments on sample size (Figure 9). However, our observation is that higher local density generally reduces the likelihood of the algorithm misclassifying a lower-dimensional manifold as a higher-dimensional one. When points from different manifolds are located close together. the algorithm may incorrectly treat them as part of the same manifold, leading to an overestimation of intrinsic dimensionality. In this experiment, our goal was to show that different algorithms vary in the distance which this misclassification occurs - the smaller this distance, the better the algorithm.
Practical guidance for LID estimation In practice, for low-dimensional manifolds recommend using ESS, as it is fast and stable, though it becomes biased in higher dimensions. If the datasets are higher-dimensional and one has more computational resources, we recommend using NB. It showed the best stability and performance and is not biased in higher dimensions. Although the training time not much longer than for other neural algorithms, the prediction time is an order of magnitude higher than, e.g., LIDL, but the performance is much better than any other algorithm we tested.
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H
SOME OTHER RESULTS FOR LIDL AND FLIPD FROM THE COMPARISON
Mean LID estimated by LIDL for diffetent values of
1000
Spiral (IDR)
193.2
38.1
26.677
1.9763
5.7054
2.1493
-14.271
Uniform (IDR)
89.149
42.094
33.343
34.703
29.495
28.05
23.904
800
Funnel (IDR)
27.524
27.831
19.581
14.176
18.466
12.414
-2.7025
-600
benchmark
Moon (IDR)
26.38
20.263
22.344
12.314
23.193
14.268
-2,9294
Gaussians (IDR)-
86.434
22.523
15.501
17.243
25.48
14.003
-00443222
Spaghetti
31.989
1.3064
-3.922
200
Spheres (IDR)
40.56
19.827
20.471
17.899
12.726
18.274
11.025
T22100 D 9STO'0 8200'0
Le290 o 21800 9900
0625 0.125
0.23
0.3
03 1.1
0.23
00000
0.125
1028
(0.125
10000 0
10.0623
[0.03322
10.
dellas
Figure 25: On this plot we present how values of average LID estimate changes for different values of 8 parameter and different IDR datasets for LIDI algorithm.
1 USE OF LARGE LANGUAGE MODELS (LLMS)
LLMs used only for light copy-editing (e.g., grammar and phrasing) after the technical content was written by the authors. All content was verified by the authors.
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Mean LID estimated LIDL (original manifold) for diffetent values
1000
Spiral (IDR)
1.5702
1.7905
1.5679
0.84964
0.40249
0.11589
0.20674
Uniform (IDR)
20.431
20.463
20.265
19.569
19.853
19.889
18.623
800
Punnel (IDR)
2.3321
2.3842
1,9051
1.8724
1.6193
1.6662
1.4879
benchmark
Moon (IDR)
3.0303
3.0038
3.0997
2.9651
2.9703
2.7362
2.1749
-400
Gaussians
3.4536
4.805
4.5762
3.0489
2.245
Spaghetti (IDR)
28.127
24.596
7,0889
3.2979
2. 2.8883
1.7011
1.7441
200
Spheres (IDR)
6.4993
6.1384
5.1957
4.6907
4.0347
3.5117
2.905
(0.0039 0.0788 0.01566]
0.125
12.125
0.25
0.5
(02500 0.5
0.23
(0.0312 0,0225
[0.0625 0.125
200000 2015
[0.125
dellas
Figure 26: On this plot we present how values of average LID estimate changes for different values of
ò parameter and different datasets for LIDL algorithm. Those datasets made of original manifold coordinates before IDR transformation. Ambient space of those datasets 30.
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Mean LID estimated by LIDL (original manifold padding) diffetent values
Uniform (IDRV
14.653
8.7016
18.976
32,462
32.719
7.3864
0.10112
800
Funnel (IDI)
-107.58
-51.015
10.798
11.196
-2.72-13
-0.0026183
2.7308
600
Moon (IDF)
16.361
61.844
82.918
21.308
42.445
77.2 77.2993
35 35.837
benchhmark
400
(IDR)
6.0647
3.2346
Spaghetti (IDR)-
43.883
1.9655
9,8484
36.254
9.4618
19.82
12.134
Spheres (IDR)
11.149
7.3551
4,6377
27.493
11.822
-1.0625
17.812
0803 0.000 0077 0.01 16656
"I s0
PSTO'O ECESO
0.25
(0:10) 1255000 0223 0.5
10.0312 0.0825
0.125
(0.0625
820000
dellas
Figure 27: On this plot we present how values of average LID estimate changes for different values of d parameter and different datasets for LIDL algorithm. Those datasets made of original manifold coordinates before IDR transformation padded with 0 to be of higher dimension. Ambient space of those datasets is 784.
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Mean LID estimated by LIDL for diffetent values of 6
1000
FMNIST (100.00%)
109.78
FMNIST (50.00%)
564 m1
NI9.1
511 K5
063 17
365 31
241 70
11604
FMNIST (25.00%)
583.15
358,82
13882
FMNIST (12.50%)
782.59
590.14
426.23
167
800
FMNIST (6. (5%)
$26.84
103 15
764.04
697.46
882.4
322.7
216.30
FMNIST (3. 13))
773
299-14
FMNIST (1.56%)
781.5
972.23
89.675
FMNIST (0. 78%)
766.49
780 49
791.07
761 96
737.19
445.24
54.77 74.724
600
benchmark
FMNIST (0.39%)
774.4
793.18
774,02
965-25
121.34
FMNIST (0.20%)
768.47
700.47
818.35
814.96
220.06
FMNIST (0.10%)
747.75
775 85
281.3
805 75
801
656 64
320.5
FMNIST (0.05%)
803.76
736.55
866.83
309.12
400
FMNIST (0.02%)
749.73
770.9
802.37
867.11
562.06
FMNIST (downscaled)
188.06
174.04
160.96
138.73
101.99
00.886
37.77
FMNIST (add dim, +dd)
$22.07
282
227.01
1
126.68
FMNIST (add dim, +4d)
153.26
FMNIST (add dim, 8d)
542 m3
415.75
317
257.38
118
136.56
2016
200
FMNIST (stretched 25)
226
172.87
1433.05
$2.075
35.200
21.2
FMNIST (stretched x*)
128.53
108.83
54.687
34.004
FMNIST (upscaled)
469 115
353
270.007
197.11
122.74
27.385
16. 754
2oooo 101500
0.0312|
0.0625]
0.125
0.25
0.5
10.25 1.
007788800 0.01158
0.0312
[0.0312 0.0625
0.125
0.25
[0.125
10.0625
(oo00 100000 to
100007)
(001150
deltas
Figure 28: On this plot present how values of average LID estimate changes for different values
of parameter and different modified FMNIST datasets for LIDL algorithm.
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Maan LTD aetimated RUPP for diffebent walow
Syral J0R
40-16
VVB
12-14
20.00
28.004
27:41
26.77
21.441
103
892.2
325-75
19672
36.46
30513
15.11
27797
4.70122
20.00
24611
27.012
372277
26.06
21.400
401
231.27
120.42
61
-2.35
M4.449
36.712
15.555
1733
20,00
14.17
24:11
25.00
22.316
26.11
18:04
20622
4440
11
73.00
W1
11
24.90
74.14
Figure 29: On this plot we present how values of average LID estimate changes for different values of parameter and different IDR datasets for FLIPD algorithm. To improve clarity, we present results for every 4th t.
Mean LID estimated by FLIPD for diffetent values
10000
PNINSTT (100 00%0
FMNEST (50 (0%)
FMNEST (12,50%)
-00
IMNIST (6.7570
611.20
10.2
CMC
STAIO
3
PMNIST (3 12%)
FNNIST (.56%)
IMNIST (0.70%)
FMNNSTT (0 (0.29%)
41312
11ke
10.24
T14
21
15:544
PMINIST (0 20%)
04.40
422.4
60
2:37
ENNIST (0.10%)
400
FMINIST 0.0229)
40
PMNIST (downscaled)
PMNNIS Tadd +0d)
1286
1962
2019
200
TMNIST (wdd
TMNIST (steecched 52
2616
IEM
31357
1
FMNISU stretched 2
2724
7LBV
(aperaled)
0.0
1.059778NNE 06
000024441 4400255
D4409 PESOOOO
1996
10000
004081 A17224
Figure 30: On this plot we present how values of average LID estimate changes for different values of parameter and different modified FMNIST datasets for FLIPD algorithm. To improve clarity, we present results for every 4th l.
34


Reviews:
[{"review_id": "rDdJyNLYA6", "reviewer": "Reviewer_5JNw", "summary": "This work is seeks to bridge a gap in common benchmarking approaches for LID estimators: the literature typically focuses on either (1) simple manifolds of known dimension or (2) realistic manifolds of unknown dimension. To address this gap, a toolkit is proposed for designing new LID benchmarks whose (1) geometric properties are understood but which (2) resemble realistic data. Foremost among these tools is inverse domain representation (IDR), wherein a simple, well-understood manifold $M$ is embedded isometrically into the space of a real-world dataset $X$ using the principal components of $X$. In the paper, this amounts to creating images that *look* like FMNIST but have the geometric properties of spheres, curves, solid hypercubes, etc. Other tools are also introduced, e.g. warping the manifold or ways of adding ambient dimensions.\n\nThe work constructs a sequence of datasets by applying these tools to simple known manifolds with geometric properties that are pathological (at least from the perspective of common LID estimators). Four estimators are compared on these datasets - expected simplex skewness (ESS), normal bundle (NB), LIDL, and FLIPD. These estimators, and especially the latter 3 neural net-based ones, underperform compared to results on simpler datasets in their respective papers, establishing this set of benchmarks as a potential new frontier for LID estimation.", "questions": "- See weaknesses 1 and 2. Do you have any insights on why the individual estimators perform as they do in specific scenarios and how they can be improved?\n- Can you expand on these phrases and perhaps provide a specific citation if appropriate? I had understood density constancy as a requirement more for classical LID estimators than newer neural ones like NB, LIDL, FLIPD.\n\t- \"in LIDL this bias is a function of the laplacian of the density\"\n\t- \"existing algorithms make specific assumptions during derivation -- [...] local density constancy\"", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- In my opinion, this work does what it sets out to do. It goes a decent way towards bridging the aforementioned gap between simple LID benchmarks and real-world datasets.\n- It is also clearly written.\n- Its strength is in the way it systematizes methods for creating benchmarks and applies them to create new ones. Its \"toolbox\" is clever and original. If it gains traction, I could see it forming a new de facto leaderboard for LID estimators.\n- Its main weakness is when it starts to work with individual estimators - see weaknesses for thorough feedback on this.\n\nNevertheless, as I said, this work achieves what it sets out to do, and I recommend it for acceptance.", "weaknesses": "Much of this work ends up being a direct comparison of 4 popular LID estimators. I think the work's main weak spots are in the specifics of this comparison:\n- The work only summarizes at a high-level the results of its benchmarks, and provides almost no analysis of *why* specific LID estimators do well or poorly on the constructed benchmarks. The community is left by themselves to figure out what exactly is going on with the individual estimators. I believe this work would be more impactful if it were to propose fewer datasets but provide more explanation of the failure modes of specific estimators on each.\n- Pursuant to the previous point, this work suggests no improvements nor provides clear next steps for improving the state of the art. Having experimented with these methods on a vast suite of LID estimation problems, the authors should be in a good position to recommend solutions to the highlighted failure modes.\n- In my opinion, there is some nuance missing in the way the work compares estimators:\n\t- Efficiency goes ignored in this analysis. My understanding is that, if we were to choose the fastest possible usable hyperparameters for each method and amortize training time, FLIPD > LIDL >> NB >> ESS in terms of runtime cost. This is approximately the opposite order to the ranking in this manuscript.\n\t- The discussion work mostly treats these estimators as drop-in replacements for each other when in fact they each have advantages beyond error and efficiency. In particular, NB/LIDL/FLIPD are useful (1) online, for new points and (2) when the model's own LID is of interest. Some more specific advantages:\n\t\t- NB supplies normal vectors of the manifold, which have utility beyond a simple LID number.\n\t\t- LIDL is the only of these that works for normalizing flows and density estimators in general.\n\t\t- The FLIPD estimate of a point is differentiable and can be estimated in a single forward pass of its model.\n\t\tIf perfect LIDs were my only consideration, ESS would always be my default choice as long as it is tractable (but it is often not).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "kmAGIt3Gij", "reviewer": "Reviewer_Sef9", "summary": "The paper analyzes algorithms for estimating local intrinsic dimension, focusing on ESS, NB, LIDL, and FLIPD. The experiments cover simple synthetic datasets with known dimension and realistic datasets with complex structure and unknown dimension. For the realistic case, the paper applies transformations that preserve manifold dimension or change it in a known way and then measures changes in reported dimension to evaluate robustness. The study investigates controlled factors that affect LID estimates, including manifold curvature, thickness, sample size, non uniform density, and proximity to boundaries.", "questions": "- What is the number of samples used in the Gaussian IDR experiment, and is it sufficient for reliable estimation in the tails? What are the number of samples used for the other experiments?\n\n- Can the authors define precisely distance to the neighbor on the same manifold versus distance to the nearest neighbor in the ambient space, and explain how these distances are instantiated in the Funnel and Spiral experiments?\n\n- In line 1471, it is claimed that \"Arrows (MS) has many V-shaped corners as artifact of translating and rotating\". Can this be further explained?\n\n- When two arrows overlap it is claimed that RGB values are added (Line 1471), but this effect is not visible in Figure 2. Can the authors clarify this?\n\n- Minor typos:\n\n  - Line 141: \"Sec. 1\" should be \"Sec. 2.\"\n\n  - Line 168: It is unclear whether the manifold is S^4 or embedded in a 6 dimensional space. Please clarify the intended statement.\n\n  - Line 242: The set description contains typos. Please correct the notation.\n\n  - Line 312: \"algorithm error\" should be \"algorithm's error.\"\n\n  - Line 317: \"Figure 8\" refers to another paper but links to the Figure 8 in the current paper.\n\n  - Line 412: \"too big our computational\" should be \"too big for our computational.\"\n\n  - Line 445: \"worse to\" should be \"worse than.\"\n\n  - Line 455: The word \"while\" should be removed.\n\n  - Figure references are inconsistent, alternating between \"Fig.\" and \"Figure\" throughout the paper.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The experimental design explicitly probes factors known to affect LID estimates, including curvature, thickness, sample size, non uniform density, and proximity to boundaries.\n\n- The experimental section is thorough.\n\n- The empirical design spans simple known dimension synthetic data and complex unknown dimension real data, which provides a broad stress test for the methods.\n\n- The paper systematically highlights shortcomings of existing LID methods and existing benchmarks through controlled experiments.", "weaknesses": "- The Gaussian (IDR) experiment does not report the number of samples used, and tail regions such as standard deviation beyond 2 standard deviations require high sample sizes to get enough samples in the region to have stable estimates, especially for model trained methods like FLIPD and LIDL. The number of training and test samples should be reported across other experiments as well.\n\n- The Funnel (IDR) experiment is placed under nearby manifolds, but as described it is a single component and seems more relevant to thin manifolds.\n\n- The Spiral (IDR) experiment under nearby manifolds appears to have extremely low point density on the outer spiral, which can cause any algorithm to fail.\n\n- Section 3.5 nearby manifolds uses terms like distance to the neighbor on the same manifold versus distance to the nearest neighbor in the ambient space without a precise definition, and the Funnel and Spiral experiments do not make these distances clear.\n\n- Figure 9 uses a log x axis and allocates a large fraction of the range to fewer than 100 samples, which is hard to interpret for FMNIST where the ambient dimension is 28*28.\n\n- Figure 23 (d) would be clearer with the x axis limited to a lower range because the current figure is unreadable.\n\n- The paper mentions using PyTorch interpolation for resizing in Upscaled ASE and possibly other experiments, and known aliasing artifacts in this method might affect LID [1].\n\n- Section 2 mentions audio but there are no audio experiments, and removing the audio discussion would improve focus.\n\n- Section 3.7 cites work showing ESS is invariant to sample size for artificial datasets, but comparable results for the other algorithms on synthetic data where ground truth is known are missing.\n\n[1] Parmar, Gaurav, Richard Zhang, and Jun-Yan Zhu. \"On aliased resizing and surprising subtleties in gan evaluation.\" Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2022.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "kWdlFWP6lB", "reviewer": "Reviewer_iGTt", "summary": "The paper argues that current evaluations of local intrinsic dimension (LID) estimators are either too simple (synthetics with known LID) or unverifiable (real data with unknown LID). It proposes a benchmarking framework that: (i) maps the same manifold across domains (e.g., to images/audio) to test architecture/inductive-bias invariance; (ii) hardens existing datasets along specific geometric/measure aspects (non-uniform density, curvature, boundaries, nearby/thin manifolds); and (iii) introduces controlled real-data transformations (monotonic warps, ambient-dimension extension, auxiliary dimension injection) that induce known LID shifts or preserve LID, enabling relative ground-truth checks on real data. Across a suite of synthetic, “real-like,” and transformed FMNIST datasets, the authors show that several recent neural LID estimators (LIDL, FLIPD, NB) exhibit systematic failures under these stressors, whereas a classic method (ESS) is often more stable on low-dimensional cases; none is universally robust.", "questions": "1. For ME/ASE/ADI, can you provide proof sketches or lemmas that (a) ME preserves LID; (b) ASE preserves LID; (c) ADI increases LID by the number of injected independent parameters plus synthetic verifications before FMNIST?\n\n2. Beyond the PCA argument, can you report quantitative diagnostics (isometry error, curvature change) showing IDR preserves manifold geometry for your choices; and try one alternate basis (e.g., DCT) to check the invariance of conclusions?\n\n3. Could you include one audio benchmark (e.g., monotonic resampling—ASE; dynamic range warp—ME) to substantiate the “any continuous domain” claim?\n\n4. Will you add per-aspect scalar metrics with 95% CIs (e.g., MAE/ΔLID error, invariance score) to complement plots, and report seeds and tuning protocols for each method?\n\n5. Arrows seems especially challenging; can you ablate which property (corners, occlusion, color composition) breaks each method, and whether smoothing/anti-aliasing mitigates it?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Originality.\n\nIntroduces a domain-mapping toolbox (IDR) to test the same manifold across architectures/domains—surfacing inductive-bias sensitivity that simple synthetics miss.\n\nDesigns targeted stressors (non-uniform densities, curvature, edges, nearby/thin manifolds) and controlled real-data transforms (ME/ASE/ADI) to create falsifiable checks (LID preserved or shifted by a known amount).\n\nQuality.\n\nBroad, well-motivated experimental suite: Gaussians/Spheres/Uniform/Spaghetti, Moon/Funnel/Spiral (nearby/thin), Arrows (real-like with known LID), plus FMNIST transforms; summarizes per-aspect outcomes in tables/figures.\n\nIdentifies concrete failure modes (e.g., NB overestimation on Gaussians/Spheres; LIDL/FLIPD instability; sample-size bias).\n\nClarity.\n\nClear “what/why” in Introduction and contributions; formalization of IDR (PCA-based embedding that preserves manifold geometry for the tests) and per-transform rationale are well explained; figures are readable and tied to claims.\n\nSignificance.\n\nBenchmarks uncover architecture/domain sensitivity and lack of invariance that prior evaluations hide; they provide a useful bar for future LID methods and highlight where current neural estimators fail (e.g., Arrows, boundaries, size-dependence).", "weaknesses": "1. Despite the stated “any continuous domain” framing, experiments are concentrated in the image domain (FMNIST, Arrows) and four LID estimators. This limits generality of the conclusions about cross-domain invariance. Add at least one non-image domain (audio/EEG) to exercise ME/ASE/ADI beyond images.\n\n2. For ME/ASE/ADI, the paper argues LID preservation/shift qualitatively; however, formal guarantees (and error bounds under finite samples) are not given. Tighten the theory or at least provide simulation checks where the induced ΔLID is analytically known before porting to FMNIST.\n\n3. IDR relies on PCA directions from a single FMNIST class chosen for visual coherence; the Discussion acknowledges this choice and compute constraints. This may bias domain mappings and downstream conclusions about architecture invariance. Consider validating IDR with alternative bases (e.g., DCT) or multiple classes, and quantify any distortions with pairwise-distance/curvature diagnostics.\n\n4. The authors note that more principled metrics per aspect are “future work.” Even simple aggregated scores (MAE vs. known LID; calibration of ΔLID; invariance indices) with confidence intervals across seeds would make comparisons more decisive.\n\n5. Several methods are sensitive (e.g., ESS neighbors; LIDL δ; FLIPD t). The paper shows this qualitatively, but a uniform protocol (grid, selection rule, robustness bands) would avoid accusations of cherry-picking and clarify practical deployability.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "Qq8Qnhy3ck", "reviewer": "Reviewer_QG1L", "summary": "This paper proposes several methods for generating high-dimensional datasets with known intrinsic dimensions from various perspectives, for the purpose of evaluating local intrinsic dimension (LID) estimation methods, and presents examples of how these datasets can be used to assess the performance of different LID estimation algorithms.", "questions": "As pointed out in the “Weaknesses” section, the paper should clearly articulate why (local) intrinsic dimension estimation is important in the first place.\n\nIn addition, to convey the significance of the proposed benchmark datasets, it is necessary to explain why neural network–based methods are important, and simultaneously, to discuss the limitations of traditional fractal-dimension approaches such as box-counting dimension.\n\nSpecifically, please describe what the limitations of non–neural-based methods are, and clarify why the proposed benchmark datasets are designed primarily for evaluating neural network–based methods.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 3, "strengths": "The paper proposes several methods for generating datasets that are well-suited or particularly challenging for intrinsic dimension estimation from various perspectives, and it conducts comprehensive evaluation experiments on representative (neural) LID estimation methods.\n\nThe diverse data manifolds presented in Subsections 3.1 through 3.9 each represent scenarios where successful intrinsic dimension estimation is desirable, and as such, the proposed benchmarks provide a valuable contribution in their current form.", "weaknesses": "The paper emphasizes the importance of benchmarking LID estimation and proposes various methods for constructing datasets; however, it lacks a discussion that would convince the broad ICLR audience why (local) intrinsic dimension estimation is important and to what extent it is a significant problem in the first place.\n\nWhile the procedures for creating benchmark datasets are described in sufficient detail, the discussion of LID estimation methods themselves is insufficient. The appendix provides only brief, one-paragraph summaries of individual methods, but it does not offer insights into which methods succeed or fail on each benchmark dataset, and why, which would be necessary to derive meaningful understanding from the experiments.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)