Intrinsic Dimension Estimation Using Wasserstein Distances

Recently, practical applications of machine learning involve a very large number of features, often many more than there are samples on which to train a model. Despite this imbalance, many modern machine learning models work astonishingly well. One of the more compelling explanations for this behavior is the manifold hypothesis, which posits that, though the data appear to the practitioner in a high-dimensional, ambient space, $\mathbb{R}^{D}$, they really lie on (or close to) a low dimensional space $M$ of “dimension” $d\ll D$, where we define dimension formally below. A good example to keep in mind is that of image data: each of thousands of pixels corresponds to three dimensions, but we expect that real images have some inherent structure that limits the true number of degrees of freedom in a realistic picture. This phenomenon has been thoroughly explored over the years, beginning with the linear case and moving into the more general, nonlinear regime, with such works as Niyogi et al., (2008, 2011); Belkin & Niyogi, (2001); Bickel et al., (2007); Levina & Bickel, (2004); Kpotufe, (2011); Kpotufe & Dasgupta, (2012); Kpotufe & Garg, (2013); Weed et al., (2019); Tenenbaum et al., (2000); Bernstein et al., (2000); Kim et al., (2019); Farahmand et al., (2007), among many, many others. Some authors have focused on finding representations for these lower dimensional sets (Niyogi et al.,, 2008; Belkin & Niyogi,, 2001; Tenenbaum et al.,, 2000; Roweis & Saul,, 2000; Donoho & Grimes,, 2003), while other works have focused on leveraging the low dimensionality into statistically efficient estimators (Bickel et al.,, 2007; Kpotufe,, 2011; Nakada & Imaizumi,, 2020; Kpotufe & Dasgupta,, 2012; Kpotufe & Garg,, 2013; Ashlagi et al.,, 2021).

In this work, our primary focus is on estimating the intrinsic dimension. To see why this is an important question, note that the local estimators of Bickel et al., (2007); Kpotufe, (2011); Kpotufe & Garg, (2013) and the neural network architecture of Nakada & Imaizumi, (2020) all depend in some way on the intrinsic dimension. As noted in Levina & Bickel, (2004), while a practitioner may simply apply cross-validation to select the optimal hyperparameters, this can be very costly unless the hyperparameters have a restricted range; thus, an estimate of intrinsic dimension is critical in actually applying the above works. In addition, for manifold learning, where the goal is to construct a representation of the data manifold in a lower dimensional space, the intrinsic dimension is a key parameter in many of the most popular methods (Tenenbaum et al.,, 2000; Belkin & Niyogi,, 2001; Donoho & Grimes,, 2003; Roweis & Saul,, 2000).

We propose a new estimator, based on distances between probability distributions, as well as provide rigorous, finite sample guarantees for the quality of the novel procedure. Recall that if $\mu,\nu$ are two measures on a metric space $(M,d_{M})$, then the Wasserstein-$p$ distance between $\mu$ and $\nu$ is

where $\Gamma(\mu,\nu)$ is the set of all couplings of the two measures. If $M\subset\mathbb{R}^{D}$, then there are two natural metrics to put on $M$: one is simply the restriction of the Euclidean metric to $M$ while the other is the geodesic metric in $M$, i.e., the minimal length of a curve in $M$ that joins the points under consideration. In the sequel, if the metric is simply the Euclidean metric, we leave the Wasserstein distance unadorned to distinguish it from the intrinsic metric. For a thorough treatment of such distances, see Villani, (2008). We recall that the Hölder integral probability metric (Hölder IPM) is given by

where $C_{B}^{\beta}(\Omega)$ is the Hölder ball defined in the sequel. When $p=\beta=1$, the classical result of Kantorovich-Rubinstein says that the Wasserstein and Hölder distances agree. It has been known at least since Dudley, (1969) that if a space $M$ has dimension $d$, $\mathbb{P}$ is a measure with support $M$, and $P_{n}$ is the empirical measure of $n$ independent samples drawn from $\mathbb{P}$, then $W_{1}^{M}(P_{n},\mathbb{P})\asymp n^{-\frac{1}{d}}$. More recently, Weed et al., (2019) has determined sharp rates for the convergence of this quantity for higher order Wasserstein distances in terms of the intrinsic dimension of the distribution. Below, we find sharp rates for the convergence of the empirical measure to the population measure with respect to the Hölder IPM: if $\beta<\frac{d}{2}$, then $d_{\beta}(P_{n},\mathbb{P})\asymp n^{-\frac{\beta}{d}}$ and if $\beta>\frac{d}{2}$ then $d_{\beta}(P_{n},\mathbb{P})\asymp n^{-\frac{1}{2}}$. These sharp rates are intuitive in that convergence to the population measure should only depend on the intrinsic complexity (i.e. dimension) without reference to the possibly much larger ambient dimension.

The above convergence results are nice theoretical insights, but they have practical value, too. The results of Dudley, (1969); Weed et al., (2019), as well as our results on the rate of convergence of the Hölder IPM, present a natural way to estimate the intrinsic dimension: take two independent samples, $P_{n},P_{\alpha n}$ from $\mathbb{P}$ and consider the ratio of $W_{p}^{M}(P_{n},\mathbb{P})/W_{p}^{M}(P_{\alpha n},\mathbb{P})$ or $d_{\beta}(P_{n},\mathbb{P})/d_{\beta}(P_{\alpha n},\mathbb{P})$; as $n\to\infty$, the first ratio should be about $\alpha^{d}$, while the second should be about $\alpha^{\frac{\beta}{d}}$, and so $d$ can be computed by taking the logarithm with respect to $\alpha$. The first problem with this idea is that we do not know $\mathbb{P}$; to address this, we instead compute the ratios using two independent samples. A more serious issue regards how large $n$ must be in order for the asymptotic regime to apply. As we shall see below, the answer depends on the geometry of the supporting manifold.

We define two estimators: one using the intrinsic distance and the other using Euclidean distance

where the primes indicate independent samples of the same size and $G$ is a graph-based metric that approximates the intrinsic metric. Before we go into the details, we give an informal statement of our main theorem, which provides finite sample, non-asymptotic guarantees on the quality of the estimator¹¹1Explicit constants are given in the formal statement of Theorem 22:

Although the guarantees for $d_{n}$ and $\widetilde{d_{n}}$ are similar, empirically $\widetilde{d}_{n}$ is much better, as explained below. Note that the ambient dimension $D$ never enters the statistical complexity given above. While the exponential dependence on the intrinsic dimension $d$ is unfortunate, it is likely necessary as described below.

While the reach, $\tau$, determines the sample complexity of our dimension estimator, consideration of the injectivity radius, $\iota$, is relevant for practical application. Both geometric quantities are defined formally in the following section, but, to understand the intuition, note that, as discussed above, there are two natural metrics we could be placing on $M=\operatorname{supp}\mathbb{P}$, the Euclidean metric and the geodesic distance. The reach is, intuitively, the size of the largest ball with respect to the ambient metric such that we can treat points in $M$ as if they were simply in Euclidean space; the injectivity radius is similar, except it treats neighborhoods with respect to the intrinsic metric. Considering that manifold distances are always at least as large as Euclidean distances, it is unsurprising that $\tau\lesssim\iota$. Getting back to dimension estimation, specializing to the case of $\beta=p=1$, and recalling (2), there are now two choices for our dimension estimator: we could use Wasserstein distance with respect to the Euclidean metric or Wasserstein distance with respect to the intrinsic metric (which we will denote by $W_{1}^{M}$). We will see that if $\iota\approx\tau$, then the two estimators induced by each of these distances behave similarly, but when $\iota\gg\tau$, the latter is better. While we wish to use $W_{1}^{M}(P_{n},P_{n}^{\prime})$ to estimate the dimension, we do not know the intrinsic metric. As such, we use the $k$NN graph to approximate this intrinsic metric and introduce the measure $W_{1}^{G}(P_{n},P_{n}^{\prime})$. Note that if we had oracle access to geodesic distance $d_{M}$, then the $W_{1}^{M}$-based estimator $\widetilde{d}_{n}$ would only require $\asymp\iota^{-d}$ samples. However, our $k$NN estimator of $d_{M}$, unfortunately, still requires the $\tau^{-d}$ samples. Nevertheless, there is a practical advantage of $\widetilde{d}_{n}$ in that the metric estimator can leverage all $N=2(1+\alpha)n$ available samples, so that $\widetilde{d}_{n}$ works if $N\gtrsim\tau^{-d}$ and only $n\gtrsim\iota^{-d}$, whereas for $d_{n}$ we require $n\gtrsim\tau^{-d}$ itself.

A natural question: is this more complicated approach necessary? i.e., is $\iota\gg\tau$ on real datasets? We believe that the answer is yes. To see this, consider the case of images of the digit 7 (for example) from MNIST (LeCun & Cortes,, 2010). As a demonstration, we sample images from MNIST in datasets of size ranging in powers of 2 from $32$ to $2048$, calculate the Wasserstein distance between these two samples, and plot the resulting trend. In the right plot, we pool all of the data to estimate the manifold distances, and then use these estimated distances to compute the Wasserstein distance between the empirical distributions. In order to better compare these two approaches, we also plot the residuals to the linear fit that we expect in the asymptotic regime. Looking at Figure 1, it is clear that we are not yet in the asymptotic regime if we simply use Euclidean distances; on the other hand, the trend using the manifold distances is much more clearly linear, suggesting that the slope of the best linear fit is meaningful. Thus we see that in order to get a meaningful dimension estimate from practical data sets, we cannot simply use $W_{1}$ but must also estimate the geometry of the underlying distribution; this suggests that $\iota\gg\tau$ on this data manifold. More generally, we note that the injectivity radius, $\iota$, is intrinsic to the geometry of the manifold and thus unaffected by the imbedding; in contradistinction, the reach, $\tau$, is extrinsic and thus can be made smaller by changing the imbedding. In particular, when the obstruction to the reach being large is a “bottleneck” in the sense that the manifold is imbedded in such a way as to place distant neighborhoods of the manifold close together in Euclidean distance (see Figure 2 for an example), we may expect $\tau\ll\iota$. Intuitively, this matches the notion that the geometry of the data would be simple if we were to have access to the “correct” coordinate system and that the difficulty in understanding the geometry comes from its imbedding in the ambient space.

We emphasize that, like many estimators of intrinsic dimension, we do not claim robustness to off-manifold noise (Levina & Bickel,, 2004; Farahmand et al.,, 2007; Kim et al.,, 2019). Indeed, any “fattening” of the manifold will force any consistent estimator of intrinsic dimension to asymptotically grow to the full, ambient dimension as the number of samples grows. Various works have included off-manifold noise in different ways, often with the assumption that either the noise is known (Koltchinskii,, 2000) or the manifold is linear (Niles-Weed & Rigollet,, 2019). Methods that do not make these simplifying assumptions are often highly sensitive to scaling parameters that are required inputs in such methods as multi-scale, local SVD (Little et al.,, 2009). Extensions of our method to such noisy settings are a promising avenue of future research, particularly in understanding the effect of this noise on downstream applications as is done for Lipschitz classification in metric spaces and the resulting dimension-distortion tradeoff found in Gottlieb et al., (2016); in this work, however, we confine our theoretical study to the noiseless setting. The primary theoretical advantage of our estimator over that of Levina & Bickel, (2004); Farahmand et al., (2007) is that we do not require the stringent regularity assumptions for our nonasymptotic rates to hold. We leave it for future empirical works whether this weakening of assumptions allows for a better practical estimator on real-world data sets.

Our main contributions are as follows:

• •

  In Section 3, we introduce a new estimator of intrinsic dimension. In Theorem 22 we prove non-asymptotic bounds on the quality of the introduced estimator. Moreover, unlike the MLE estimator of Levina & Bickel, (2004) with non-asymptotic analysis in Farahmand et al., (2007), minimal regularity of the density of the population distribution is required for our guarantees and, unlike that suggested in Kim et al., (2019), our estimator is both computationally efficient and has sample complexity independent of the ambient dimension.

• •

  In the course of proving Theorem 22, we adapt the techniques of Bernstein et al., (2000) to provide new, non-asymptotic bounds on the quality of kNN distance as an estimate of intrinsic distance in Proposition 24, with explicit sample complexity in terms of the reach of the underlying space. To our knowledge, these are the first such non-asymptotic bounds.

We further note that the techniques we develop to prove the non-asymptotic bounds on our dimension estimator also serve to provide new statistical rates in learning Generative Adversarial Networks (GANs) with a Hölder discriminator class:

• •

  We prove in Theorem 25 that if $\widehat{\mu}$ is a Hölder GAN, then the distance between $\widehat{\mu}$ and $\mathbb{P}$, as measured by the Hölder IPM, is governed by rates dependent only on the intrinsic dimension of the data, independent of the ambient dimension or the dimension of the feature space. In particular, we prove in great generality that if $\mathbb{P}$ has intrinsic dimension $d$, then the rate of a Wasserstein GAN is $n^{-\frac{1}{d}}$. This improves on the recent work of Schreuder et al., (2020).

The work is presented in the order of the above listed contributions, preceded by a brief section on the geometric preliminaries and prerequisite results. We conclude the introduction by fixing notation and surveying some related work.

We outlined the intuition behind our dimension estimation in the introduction. In this section, we formally define the estimator and analyse its theoretical performance. We first apply standard empirical process theory and our complexity bounds in the previous section to upper bound the expected Hölder IPM (defined in (1)) between empirical and population distributions:

The proof uses the symmetrization and chaining technique and applies the complexity bounds of Hölder functions found above; the details can be found in Appendix E.

We now specialize to the case where $\beta=B=1$, due to the computational tractability of the resulting Wasserstein distance. Applying Kantorovich-Rubenstein duality (Kantorovich & Rubinshtein,, 1958), we see that this special case of Lemma 15 recovers the special $p=1$ case of Weed et al., (2019). From here on, we suppose that $d>2$ and our metric on distributions is $d_{1,1}=W_{1}$.

We begin by noting that if we have $2n$, independent samples from $\mathbb{P}$, then we can split them into two data sets of size $n$, and denote by $P_{n},P_{n}^{\prime}$ the empirical distributions thus generated. We then note that Lemma 15 implies that if $\operatorname{supp}\mathbb{P}\subset M$ and $M$ is of dimension $d$, then

If we were to establish a lower bound as well as concentration of $W_{1}(P_{n},P_{n}^{\prime})$ about its mean, then we could consider the following estimator. Given a data set of size $2(\alpha+1)n$, we can break the data into four samples, $P_{n},P_{n}^{\prime}$ each of size $n$ and $P_{\alpha n},P_{\alpha n}^{\prime}$ of size $\alpha n$. Then we would have

Which distance on $M$ should be used to compute the Wasserstein distance, the Euclidean metric $\left|\left|\cdot\right|\right|$ or the intrinsic metric $d_{M}(\cdot,\cdot)$? As can be guessed from Corollary 14, asymptotically, both will work, but for finite sample sizes when $\iota\gg\tau$, the latter is much better. One problem remains, however: because we are not assuming $M$ to be known, we do not have access to $d_{M}$ and thus we cannot compute the necessary Wasserstein cost. In order to get around this obstacle, we recall the graph distance induced by a $k$NN graph:

We now have two Wasserstein distances, each induced by a different metric; to mitigate confusion, we introduce the following notation:

Given sample distributions $P_{n},P_{n}^{\prime}$, we are able to compute $W_{1}(P_{n},P_{n}^{\prime})$ and $W_{1}^{G}(P_{n},P_{n}^{\prime})$ for any fixed $\varepsilon$, but not $W_{1}^{M}(P_{n},P_{n}^{\prime})$ because we are assuming that the learner does not have access to the manifold $M$. On the other hand, adapting techniques from Weed et al., (2019), we are able to provide a non-asymptotic lower bound on $W_{1}(P_{n},P_{n}^{\prime})$ and $W_{1}^{M}(P_{n},P_{n}^{\prime})$:

An easy proof, based on the techniques (Weed et al.,, 2019, Proposition 6) can be found in Appendix E. Similarly, we can apply the same proof technique as Lemma 15 to establish the following upper bound:

The full proof is in Appendix E and applies symmetrization and chaining, with an upper bound of Corollary 14. We note, as before, that a similar asymptotic rate is obtained by Weed et al., (2019) in a slightly different setting.

We noted above (3) that we required two facts to make our intuition precise. We have just shown that the first holds; we turn now to the second: concentration. To make this rigorous, we need one last technical concept: the $T_{2}$-inequality.

The reason that the $T_{2}$ inequality is useful for us is that Bobkov & Götze, (1999) tell us that such an inequality implies, and is, by Gozlan et al., (2009), equivalent to Lipschitz concentration. We note further that $W_{1}(P_{n},P_{n}^{\prime})$ is a Lipschitz function of the dataset and thus concentrates about its mean. The constant in the $T_{2}$ inequality depends on the measure $\mu$ and upper bounds for specific classes of measures are both well-known and remain an active area of research; for a more complete survey, see Bakry et al., (2014). We have the following bound:

In order to bound the $T_{2}$ constant in our case, we rely on the landmark result of Otto & Villani, (2000) that relates $c_{2}$ to another functional inequality, the log-Sobolev inequality (Bakry et al.,, 2014, Chapter 5). There are many ways to control the log-Sobolev constant in various situations, many of which are covered in Bakry et al., (2014). We use results from Wang, (1997b), which incorporate the intrinsic geometry of the distribution, as our bound. A detailed proof can be found in Appendix B. We note that many other estimates with under slightly different conditions exits, such as that in Wang, (1997a), which requires second-order control of the density of the population distribution with respect to the volume measure and the bound in Block et al., (2020), which provides control using a measure of nonconvexity. With added assumptions, we can gain much sharper control over $c_{2}$; for example, if we assume a positive lower bound on the curvature of the support, we can apply the well-known Bakry-Émery result (Bakry & Émery,, 1985) and get dimension-free bounds. As another example, if we may assume stronger contol on the curvature of $M$ beyond that guaranteed by the reach, we can remove the exponential dependence on the reach entirely. For the sake of simplicity and because we already admit an exponential dependence on the intrinsic dimension, we present only the more general bound here. We now provide a non-asymptotic bound on the quality of the estimator $\widetilde{d}_{n}$.

We note that we have not made every effort to minimize the constants in the statement above, with our emphasis being the dependence of these sample complexity bounds on the relevant geometric quantities. As an immediate consequence of Theorem 22, due to the fact that $d$ is discrete, we can control the probability of error with sufficiently many samples. We may also apply Proposition 21 to replace $c_{2}$ with our upper bound in terms of the reach.

While the appearance of $\iota$ in Theorem 22 and Corollary 23 may seem minor, it is critical for any practical estimator. While $\alpha n=\Omega\left(\tau^{-d}\right)$, we may take $n$ as small as $\Omega\left(\iota^{-d}\right)$. Thus, using $\widetilde{d}_{n}$ instead of the naive estimator $d_{n}$ allows us to leverage the entire data set in estimating the intrinsic distances, even on the small sub-samples. From the proof, it is clear that we want $\alpha$ to be as large as possible; thus if we have a total of $N$ samples, we wish to make $n$ as small as possible. If $\iota\gg\tau$ then we can make $n$ much smaller (scaling like $\iota^{-d}$) than if we were to simply use the Euclidean distance. As a result, on any data set where $\iota\gg\tau$, the sample complexity of $\widetilde{d}_{n}$ can be much smaller than that of $d_{n}$.