Quasi-Monte Carlo Methods Enable Extremely Low-Dimensional Deep Generative Models
Authors: Miles Martinez
Alex H Williams
Venue: ICLR 2026 Poster (2026)
Keywords: representation learning
quasi monte-carlo integration
generative modeling
variational autoencoder
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: We introduce a new latent variable model that allows easy exploration of scientific datasets

Abstract: This paper introduces *quasi-Monte Carlo latent variable models* (QLVMs): a class of deep generative models that are specialized for finding extremely low-dimensional and interpretable embeddings of high-dimensional datasets.
Unlike standard approaches, which rely on a learned encoder and variational lower bounds, QLVMs directly approximate the marginal likelihood by randomized quasi-Monte Carlo integration.
While this brute force approach has drawbacks in higher-dimensional spaces, we find that it excels in fitting one, two, and three dimensional deep latent variable models.
Empirical results on a range of datasets show that QLVMs consistently outperform conventional variational autoencoders (VAEs) and importance weighted autoencoders (IWAEs) with matched latent dimensionality.
The resulting embeddings enable transparent visualization and *post hoc* analyses such as nonparametric density estimation, clustering, and geodesic path computation, which are nontrivial to validate in higher-dimensional spaces.
While our approach is compute-intensive and struggles to generate fine-scale details in complex datasets, it offers a compelling solution for applications prioritizing interpretability and latent space analysis.

Published as a conference paper at ICLR 2026
QUASI MONTE CARLO METHODS ENABLE EXTREMELY LOW-DIMENSIONAL DEEP GENERATIVE MODELS
Miles Martinez 122,, Alex H. Williams
Department of Electrical and Computer Engineering, Duke University
2 Center for Cognitive Neuroscience, Duke University
3 Center for Neural Science, New York University
Center for Computational Neuroscience, Flatiron Institute
* mi miles, martinezdduke edu, o alex. h. williamsênyu edu
ABSTRACT
This paper introduces quasi-Monte Carlo latent variable models (QLVMs): class of deep generative models that are specialized for finding extremely low- dimensional and interpretable embeddings of high-dimensional datasets. Un- like standard approaches, which rely on a learned encoder and variational lower bounds, QLVMs directly approximate the marginal likelihood by randomized quasi-Monte Carlo integration. While this brute force approach has drawbacks in higher-dimensional spaces, find that it excels in fitting one, two, and three dimensional deep latent variable models. Empirical results on range of datasets show that QLVMs consistently outperform conventional variational autoencoders (VAEs) and importance weighted autoencoders (IWAEs) with matched latent di- mensionality. The resulting embeddings enable transparent visualization and post hoc analyses such as nonparametric density estimation. clustering, and geodesic path computation, which are nontrivial to validate higher-dimensional spaces. While our approach is compute-intensive and struggles generate fine-scale de- tails in complex datasets, it offers a compelling solution for applications prioritiz- ing interpretability and latent space analysis.
INTRODUCTION
Deep generative models are mainstay of contemporary machine learning. addition to synthe- sizing complex images, sounds, and sentences, many researchers have sought to use these models to identify human-interpretable representations of complex data distributions. For example, vari- ational autoencoders (VAEs) have been applied in scientific analysis of motor kinematics (Luxem et al., 2022), vocal communication patterns (Goffinet et al., 2021), 1), single cell gene expression (Ding al., 2018 8; Grønbech et al., 2020; Yan al., 2022), neural population dynamics (Pandarinath al., 2018), and extracellular spike waveforms (Beau et al., 2025).
Reducing dimensionality is seen key step towards interpretability these models. VAEs achieve this by introducing a bottleneck layer. The size of this layer is hyperparameter and often is chosen to be modestly large. For VAES, typical values construction quality is reported to drop off for networks with fewer than - 10 dimensions (Mai Ngoc & Hwang, 2020). As result, it is virtually universal practice to employ further dimensionality reduction methods like UMAP or t-SNE visualize the latent space of a VAE. Many papers also apply analyses such as clustering to identify e.g., Goffinet et al., 2021; Luxem et al., 2022; Peterson et al., 2024; Koukuntla et al., 2025). Even in moderately high dimensions, these post hoc analyses are non-trivial to tune and validate.
in the range of 32- latent dimensions and re-
inter--itterretabble structure in the latent space (see
Here, we demonstrate that in many cases two three dimensional latent spaces are sufficient for performant deep generative models. Contrary to conventional wisdom (see, e.g., Doersch, 2016), we find that numerical integration can be viable strategy to marginalize over the latent variables. We use randomized lattice integration rules (Dick et al., 2022) to approximate the marginal likelihood. We call the resulting model quasi-Monte Carlo latent variable model, QLVM. Although

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method requires multiple passes through the decoder each training iteration, it completely elim- inates variational upproximations that can be challenging to tune and optimize (see Rainforth et al., 2018: He et al., 2019; Dai et al., 2020; Kinoshita et al., 2023, and Fig. 3 in this paper). Ultimately, we find that this approach decisively outperforms VAFs with comparable architectures and can even rival the performance of VAEs with higher capacity latent spaces.
Despite the limitations of our approach (see $4.2), there are many advantages to learning extremely low-dimensional embeddings of nigh-dimensional datasets. We can, for example, use kernel density estimators-whiih suffer a curse of dimensionality in higher dimensions- precisely characterize the aggregate posterior density over the latent space. We can also exhaustively characterize the smoothness of the embedding by querying the Jacobian of the decoder network over a dense grid of points. We demonstrate how this information can used to aid clustering analyses in $3.3, but the potential applications extend much more broadly to other topological analyses (Wasserman, 2018). Altogether, our results highlight a simple, but widely overlooked, approach to building extremely low-dimensional deep latent variable models for exploratory analysis.
2 METHODOLOGY
2.1
BACKGROUND ON DEEP LATENT VARIABLE MODELS
Let æ; E R for {l..... , denote a sequence high-dimensional, observed data points. Deep latent variable models seek to explain such datasets through the following probabilistic model:
(1) independently for i E {1, (I,...,h n). Here, a d-dimensional latent variable vector z; is drawn from a prior distribution p(z). The likelihood of each observation is parameterized by deep network, called the decoder, with trainable weights 0. We use fo(z;) E|æ; z.] to denote the mean output of the decoder. Ideally, the decoder weights would be trained to maximize the marginal likelihood:
I, ~ (a zi)
where
2i ~ p(z)
Po (e
Enn)
II po po = II po po 2z)p(z;)dz;
(2)
Computing the marginal likelihood of a datapoint requires integrating over all possible values of z;- This integration is challenging in high dimensions and the simple-minded Monte Carlo estimate,
/ po(xx x1)p(zz,dzz)
p=(x))= =
7 Σ Pa
2, 2;
(3)
j=1
where .I.....2, 2m are independent samples from p(z). is therefore typically dismissed (Doersch, 2016; Mattei Frellsen, 2018; Kingma et al., 2019). However, the quality of this approximation only deteriorates in proportion to the variance of pa (z z) for p(z) (see Owen, 2013). Thus, for sufficiently simple datasets, it is plausible that eq. (3) may suffice as criterion to train 0. To our knowledge, this possibility has not been seriously explored in deep latent variable models.
Since computing (x.) is often difficult, Variational Autoencoders (VAEs; Kingma & Welling 2014) instead target lower bound on log (z.)). To do this, VAEs introduce a second deep neural network with parameters @ called the encoder. For each datapoint the encoder outputs a conditional distribution (2 æz), which acts variational approximation to the posterior distribution of 2; given x, As a training objective, one considers:
1 Σ
FLBO (0, x)
log po(x;; 1zz -D(gs ax)) || p(=))
(4)
where 2.....,2. Zm are independent samples from (0(z xx)) and D(-II.) denotes the Kullback- Leibler (KL) divergence. In the limit that m + co, this converges to a lower bound on log Po (x,) called the evidence lower bound (ELBO). Typically, m = sample is used to obtain an unbiased estimate of the lower bound. For further background see Doersch (2016); Kingma et al. (2019).
FFllowing common practice in this literature, we use the same notation for both the probability distribution of random variable and its associated density function. We also refer the two interchangeably.
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(A) On each batch, latent samples are generated by uniformly shifting lattice of M points (Batch 1 grid and batch B grid denote examples of samples that may appear during training). (B) These samples are fed through a shared decoder, Sa, to produce reconstructed datapoints (e.g. MNIST digits). During training. log-sum-exp (LSE) reduction operator is applied to a vector of log probabilities, logp = log p(æ; 2z). to calculate eq. (5) up to a constant.
Figure 1:
(C) To approximate the latent posterior, p( xz), the same vector of log probabilities is normalized to form discrete approximation over the m lattice points. This employs Bayes' rule as described in the main text.
2.2 QUASI-MONTE CARLO LATENT VARIABLE MODELS (QLVMS)
VAEs target a lower bound on the marginal log likelihood, which is only tight when go æ;)) matches the true posterior distribution of z given æs Commonly, qa is assumed to follow simple form-e-g. Gaussian with diagonal covariance. In very low-dimensional latent spaces, we will see that this assumption can be violated and, even if it approximately holds, the true posterior can be so narrow that it is difficult for the encoder network to learn (see Fig. 3).
We therefore sought to understand whether the simple Monte Carlo approach shown in eq. (3) can be practical in situations of interest. By taking logarithms on both sides of eq. (3), we obtain:
m Σ
Mc (0) = log (x..) = log
pa(a | Σ,)
(5)
m
where æ; is any datapoint and 2....., _., are sampled points in the latent space. This is lower bound on the marginal log likelihood in expectation. Indeed, by Jensen's 's inequality:
log
Po(x, [2)]]] S log E m Σ, Po(a; ı
log po (x;)
(6)
where the expectation is taken over Σ 21,.... Zm. The inequality becomes tight Var( E, (2;; k,)] -> in this limit (see Struski et al., 2023, for a detailed analysis). Thus, we achieve a more accurate estimate of the log marginal likelihood as we increase the computational budget (i.e. sample size) of our method. Moreover, the final equality in eq. (6) holds long as the marginal distribution of each 3 is p(z). This observation motivates quasi-Monte Carlo (QMC) methods, which jointly sample the points Z1,.... : Fmm to encourage uniform spacing while preserving the appropriate marginal distribution for each sample (L' (L'Ecuyer, 2016; Owen, 2023).
co, since
To our knowledge, QMC has only been considered by handful of researchers in the context of deep generative models. Buchholz et al. (2018) applied this approach to reduce the variance of ELBO estimates-i.ee to reduce variance in eq. (4) instead of eq. (5). More recently, Andral (2024) applied randomized QMC to generative models based on normalizing flows-a class of models which is poorly suited to our motivating application of nonlinear dimensionality reduction.
In our experiments, we set p(z) 1, i.e. uniform over 10, and we sample 21,..., .m jointly as a randomly shifted lattice (Fig. 1A). For 2, we use Fibonacci lattices, which optimally tile the unit square under periodic boundary conditions (Brencis & Hinrichs, 2020). For d = 3 use Korobov lattices (see Owen, 2023, ch. 16.1). Although lattice integration rules can be applied to non-periodic functions (see Dick et al., 2022, ch. 7), focus latent spaces with periodic boundaries. To impose this desired boundary condition, we fix z + (sin 2, cos z) as the first layer of the decoder. We call the resulting model mansi-Monte Carlo latent variable model, or QLVM.
During training, we simply compute eq. (5) over the lattice, leveraging a log-sum-exp operation for numerical stability (Fig. 1B). Note that there is no need train an encoder network. At evaluation
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time, we generate samples from QLVM in the usual fashion by sampling p(z) and feeding through the decoder to evaluate (z). To project high-dimensional data into the low-dimensional embedding space, we appeal to Bayes' rule, which tells that p(z; æ;) x pa( z(), due to the uniform prior on 2;. We can therefore approximate the posterior as a normalized mixture of m Dirac masses, with weights proportional to pa (a; Σj) over lattice points (Fig. IC). One use the mean or mode of this approximate posterior as a -iw-dimnsiiona embedding of 2;.
Changing the prior distribution. As mentioned above, our experiments in $3 employ a latent space with periodic boundary conditions and a uniform prior. The uniform prior encourages points to be evenly spread and the periodic boundary conditions prevent embedded points from getting "stuck" in corners of the space. Additionally, lattice rules often have optimality properties for periodic functions (Breneis & Hinrichs, 2020), though they still work well for non-periodic functions.
It is very easy to adapt QLVMs to different sets of assumptions on the latent space. Indeed, a model uniform prior over the latent space is, in some sense, universal. Suppose that we wish to build QLVM where u denotes the latent random variable and p(u) denotes its prior. Under mild conditions, we can construct this as a one-to-one mapping u = & (=) where = is a uniform random variable and Φ-1 is the inverse CDF of p(u). Thus, to construct QLVM with prior distribution pl u). we can simply train a QLVM with a uniform prior over x, but use u = o (x) as the input to the decoder. This approach is common in the QMC literature and is often called the inversion method (see Chpt. 4 in Owen 2013 and Chpt. 8 in Leobacher & Pillichshammer 2014). In appendix E.2 show an example QLVM trained with respect to an isotropic Gaussian prior distribution on MNIST. This example model performs comparably well to QLVM with uniform prior.
2.3 RELATIONSHIP TO IMPORTANCE WEIGHTED AUTOENCODERS (IWAES)
Importance weighted autoencoders (IWAEs, Burda et al., 2016) are generalization of VAEs that target a different lower bound on the marginal log likelihood:
m
po(:;
po
(p(s)
C 6) = log
m
( x3))
j=1
where do represents the density of the proposal distribution (output by the encoder network) and ,,,...,2n Zm independent samples from this distribution. One can show that the ELBO is re- covered when m = 1. Further, so long as do places nonzero density over the latent space, becomes, in expectation, progressively tighter bound and approaches the true marginal log like- lihood as m - oc (Burda et al., 2016). Thus, IWAEs and QLVMs enjoy the same asymptotic theoretical advantages over standard VAEs. They also suffer similar computational drawbacks in both, the quality of the lower bound comes at the cost of multiple forward and backward passes through the decoder. In IWAEs, multiple backward passes through the encoder are also required.
L(3 L
IWAES are closely related to QLVMs. The major difference is that IWAES use an encoder to learn proposal distribution, Jo. while QLVMs draw samples from a fixed distribution-the prior. In this sense, QLVMs are a special case of an IWAE with fixed proposal. Below we summarize two key reasons why fixing the proposal distribution may be advantageous from a practical perspective.
Learning Good Proposals is Hard. A bad proposal can lead to a lower bound with extremely high variance (see Owen, 2013, Ch. 9). It is costly to monitor the variance of the IWAE lower bound during training and unclear how to intervene to fix problems that arise. A second, less obvious, challenge to training IWAEs was revealed by Rainforth et al. (2018), who showed that increasing vn can simultaneously improve optimization of the decoder weights, 0, while deteriorating the opti- mization of encoder weights, &b. Roughly speaking, this occurs because the IWAE bound becomes more insensitive to the proposal m oo, and the magnitude of the gradient signal is averaged away quicker than the variance is reduced. See Tucker et al. (2018); Nowozin (2018); Morningstar et al. (2021) for further discussion.
Computational Benefits of QLVMs. QLVMs avoid the need to learn an encoder network. This saves GPU memory (since encoder weights and activations don't need to be stored), computation time (since gradients don't 't need to be backpropogated through the encoder), and hyperparameter sweeps (since the encoder network and optimizer don't need to be tuned). In addition to all of
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Figure 2: (A) Marginal log likelihood estimates on heldout test data from 2D models (orange) across three datasets (QMC estimate for QLVMs, ELBO for VAE and IWAE). The QLVM bound is higher than the ELBO and IWAE bound across all 2D models. Error bars indicate 1 standard deviation across 10 random seeds. (B) Reconstructions for 2-D QLVM and VAE, with the QLVM display- ing higher-quality reconstructions. (C) Samples from the prior of 2-D QLVM, VAE, and IWAE. QLVMs show greater sample quality and sample diversity.
these benefits, training with minibatches is much more efficient with QLVMs because the randomly shifted lattice of latent points (see Fig. 1) can be re-used across all elements in the minibatch. There no simple way to re-use samples in an IWAE because, for two datapoints x and æ', , the proposal distributions (= æ)) and (0(z x') may be very different. Thus, for given batch size, 6, and total budget latent samples, 8, an IWAE will be constrained to use m = s/b samples per datapoint, whereas QLVM can utilize latent samples to estimate the loss for each datapoint.
3 RESULTS
3.1 QLVMS PERFORM FAVORABLY IN VERY LOW-DIMENSIONAL SETTINGS
As discussed in $2, prior work regards the (quasi- Monte Carlo estimator used by QLVMs as imprac- tical. Here, we show that: (a) QLVMs outperform VAES and other deep latent variable models when the latent space is constrained to be two-dimensional,
(b) QLVMs' performance is better than one might expect in this setting across a variety of non-trivial benchmarks, and (c) in cow-dimensional settings, the lower bounds targeted by standard VAEs and IWAEs are often loose.
We fit models to two image datasets (MNIST and grayscale Celeb-A), an acoustic library of bird song syllables (from Goffinet et al., 2021), and acoustic library of Mongolian gerbil vocalizations (from Peterson et al., 2024). We compared two-dimensional QLVM models to two-dimensional VAE and IWAE models. Experimental details including preprocessing and network architectures are described in $B. Importantly, we used the same decoder architecture for all models to ensure fair comparisons.
In all cases, 2D QLVMs outperformed 2D VAE and 2D IWAE models both quantitatively in terms of reconstruction error (Fig. 2A), and qualitatively on example reconstructions (Fig. 2B). Moreover, novel samples found by sampling z ~ p(z) and feeding z through the decoder were, qualitatively, more diverse and of higher quality when derived from QLVM than from VAE and IWAE models with matched latent dimensionality (Fig. 2C).
As one would expect, we found that VAEs with higher dimensional latent spaces often quantita- tively outperformed 2D Q on the most rich and challenging dataset of Celeb-A images. We show comparisons between 2D QLVMS and VAEs with various latent dimensionalities in appendix G. On simple datasets, such MNIST or the zebra finch dataset, the improvements of higher-dimennionall VAEs are qualitatively minor albeit still quantitatively detectable. In short, while 2D QLVMs struggle capture fine-scale detail in complex datasets (e.g. high resolution natural im- ages), they perform "good enough" on many other datasets interest to scientists and engineers. Thus, QLVMs an attractive method when 2D visualizations desired.
Finally, we investigated the underlying reasons why VAE and IWAE models underperform in ex- tremely low-dimensional latent spaces. We hypothesized that the encoder network struggles to iden-
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Figure 3: (A) VAE and IWAE lower bounds on marginal log likelihood (orange dots) for 2D latent spaces vs. the QMC estimate based on VAE and IWAE decoders (blue dots) from Fig. 2. Black horizontal line indicates the 2D QLVM bound the marginal log likelihood. The gap between the blue dots and the black line represents the additional benefit training with QMC samples. (B) Empirical vs. variational posteriors for MNIST, Celeb-A, and zebra finch vocalizations. Heatmap indicates 99% of probability mass the empirical posterior (estimated by a dense lattice), while the red dots and ellipses indicate means and =3 standard deviation of the variational posterior. This panel is best viewed on screen with magnification.
tify good approximation to the latent posterior in these settings, resulting in a loose bound and a poor training signal for the decoder. To assess this, we took the decoders from trained 2D VAE and 2D IWAE models and applied the quasi-Monte Carlo integration rule in eq. (5) to get tight estimate of the marginal log likelihood and the true posterior (using m = 6724 lattice points). This revealed that the bounds targeted by VAE and IWAE models were loose (Fig. 3A blue dots higher than orange, all p < 0.05 using one-sided binomial tests, Bonferroni corrected). Notably, tightening these bounds evaluation time only did immediately close the performance gap be- tween VAE/WWAE baseline decoders and the QLVM decoder (Fig. 3A blue dots below horizontal black line), demonstrating that our approach using the tighter QMC bound during training resulted in improved performance at evaluation time.
We then visualized these model posteriors as heatmaps, which qualitatively revealed that the VAE and IWAE variational approximations were often, though not always, poorly matched to the true posterior (Fig. 3B). Altogether, this analysis confirmed our suspicion that the encoder networks struggle to identify good variational approximation over very low-dimensional latent spaces, mo- tivating our approach to remove the encoder network entirely and instead use QMC to estimate the marginal log likelihood.
3.2 COMPUTATIONAL COST OF QLVMS
The quasi-Monte Carlo estimator used by QLVMs may appear to be computationally restrictive, even in low-dimensional spaces. For a fixed choice of decoder architecture, the tradeoff between QLVM performance and computational cost of training mediated by the number of lattice points, m, over the latent space. The comparable parameter for IWAES is k, the number of importance samples per data point in the batch. We therefore varied these hyperparameters and plotted model performance (negative ELBO for VAEs and IWAEs, negative QMC evidence bound for QLVMs) after 300 epochs, MNIST and zebra finch, 10 epochs, Celeb-A) against computation time (the wall time to complete an epoch of training using one NVIDIA H100 GPU). This experiment traces out ID curve that represents the Pareto front of the QLVM model class (red lines in Fig. 4). Intuitively, any competing model that falls above or to the right of this curve is suboptimal to the regardeassssss of whether computational efficiency is prioritized over reconstruction error, or vice versa. Indeed, observed that 2D VAE and 2D IWAE models with various architectures and hyperparameter settings consistently fell either above or to the right of the Pareto front, indicating underperformance.

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Figure 5: (A) Latent embeddings MNIST, colored by digit. (B) Aggregated posterior of MNIST with mean-shift centroids (red) overlaid. (C) Reconstructions of centroids in (B). (D) Smoothed Frobe- nius norm of decoder Jacobian on MNIST, with centroids from (B) overlaid. (E) Latent embeddings of gerbil vocalizations, colored by mean frequency of the vocalization. (F) Aggregated posterior of gerbil vocalizations, with mean-shile centroids (red) overlaid. (G) Reconstructions centroids from (F). (H) Smoothed log Frobenius norm of decoder Jacobian gerbil vocalizations.
3.3
USING QLVMS FOR CLUSTERING
Clustering is a notoriously challenging, yet popular. form of unsupervised analysis. Identifying clusters is particularly difficult high-dimensional spaces where conventional distance metrics (e.g. Euclidean) become less informative (Aggarwal et al., 2001). To address this, many works apply clustering algorithms on lower dimensional data representations that have been learned by a VAE or similar method (see e.g. Yan et al., 2022; Hadipour et al., 2022; Peterson al., 2024). Since VAE performance is often compromised when the latent space is very low-dimensional, this approach still requires running clustering algorithm in tens hundreds of latent dimensions. Clustering is nontrivial in this setting, leading practitioners to introduce questionable parametric assumptions on the shape of each cluster (e.g. spherical in the case of K-means clustering).

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To investigate the ability of QLVMs to help reveal clusters in digh-dimensional datasets, first examined 2D embeddings of the MNIST dataset. Using the procedure outlined in Fig. 1C, each data- point embedded into a 2D latent space and colored by the ground truth digit label (Fig. 5A). We also visualized the aggregated posterior density as a 2D histogram (Fig. 5B). By visual examina- tion, one tell the clusters in Fig. 5A do not follow simple or pre-specified shape. This motivates us to consider nonparametric clustering methods and related methods from topological data analysis (Wasserman, 2018). In Fig. 5C, we show the cluster centroids produced by the mean-shift algo- rithm (Cheng, 1995), whose locations are labeled in Fig. 5B. A single hyperparameter, bandwidth, controls the number of clusters (see appendix A.2). For illustrative purposes, we chose a relatively small bandwidth parameter to reveal finer-grained structure within each digit class.
We noticed that images with the same digit label naturally grouped together with small "gaps" separating each cluster (white space in Fig. 5A). We can show that these "gaps" correspond to regions where the map z r fz(z) defined by the decoder network changes rapidly. To reveal this, we queried the decoder Jacobian-i.e i.... the matrix containing gradients of each output dimension of fz( with respect to z. The Frobenius norm of this matrix is related to Fisher information (Berardino et al., 2017) and is measure of the discriminability of nearby points in the latent space. We find that values of x with large Jacobians present themselves at cluster boundaries (Fig. 5D), effectively highlighting the valleys of the 2D aggregate posterior in Fig. 5B. In Appendix C, we visualize latent space traversals confirming rapid changes in the decoded image. This visualization highlights a unique feature of QLVMs that is absent in methods like UMAP (which don't 't contain a generative process) as well as alternative generative models with sigher-dimensional latent spaces (where "gaps" cannot be easily identified or visualized).
We then applied clustering in the QLVM space to an acoustic library of Mongolian gerbil vocaliza- tions from Peterson et al. (2024). The number of vocal call types and the transition statisties between these calls is an open scientific question, making this dataset a more realistic and challenging testbed. We repeated the same set of analyses and visualizations as above. Embedded vocalizations segre- gated according to their mean frequency with visible "gaps" between putative clusters (Fig. 5E). The mean-shift algorithm identified a number of prototypical vocal call types (Fig. 5F-G), and (as in MNIST) the norm of the decoder's Jacobian was sharply peaked at the gaps between clusters (Fig. 5H), which corresponded to rapid changes in the decoded vocalization (appendix C). While further investigation is needed, these results suggest that the gerbil vocal repertoire can be coarsely categorized into low, middle, and high-frequency calls and that many of the finer-scale clusters (in Fig. 5G) roughly lie along spectrum. Indeed, many of the fine-scale clusters clear boundary defined by the decoder's Jacobian (Fig. 5H) and it is possible to smoothly interpolate between many, though not all, of the fine-scale vocal clusters (Figs. A4 and A5).
not separated by
3.4 USING QLVMS TO VISUALIZE CONTINUOUS LATENT DIMENSIONS
We explicitly constrain our latent space in these models be low-dimensional. However, there are cases where the frue latent dimensionality is higher than that of our model. One such case captured by the 3dShapes dataset (Burgess Kim, 2018), in which high-contrast images are generated from 6 latent factors (wall hue, floor hue, object hue, object shape, object scale, and orientation). Importantly, this dataset constructed to have minimal clustering- each latent factor, other than shape, is varied continuously. See Fig. A1 for example images from the dataset.
We trained a 2d QLVM on these data and compared the embedding to a 2d UMAP embedding (Fig. 6). UMAP artificially created small clusters of data, which are not representative of the true nature of the latent factors. QLVMs, on the other hand, represented each latent feature as a contin- uum across the latent space. Additionally, UMAP was able to separate one latent factor (floor hue) into unique clusters, but mixed all other latent factors beyond that. QLVMs smoothly encoded three of the six latent factors (wall huc, floor hue, object hue), which are responsible for the majority of variance in pixel space across images. The remaining three latent variables (shape type, scale, and orientation) were embedded in increasingly non-smooth manner. Importantly and in contrast to the experiments in $3.3, the QLVM embedding showed little evidence of clustering aggregate posterior was uniform and the decoder Jacobian showed no sharp ridges across the latent space (see
Note that the relatively uniform spread of the points across the latent space and periodic boundary condi- tions in Fig. 5 are expected due the uniform prior and our construction of periodic decoder (see $2.2).
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Object hue
Shape
Scale
Onientatio
QEVM
UMAP
Figure 6: Embeddings of the 6d 3dShapes dataset into dimensions by QLVM (top row) and UMAP (bottom row). For the QLVM, color indicates the maximum probability latent factor value that point in the space. Wall, floor, and object hues are matched the ground truth hue. Cube, cylinder, sphere, and pill are shown as blue, orange, green, and red under shape. Scale and orienta- tion range from [0, and [-30, 30] and corresponding colors range from dark blue yellow.
appendix D). Together, these results demonstrate how QLVMs can nonlinearly pack additional in- formation into a very low-dimensional space and highlight an advantage over UMAP, which is prone to hallucinate clusters (which has been previously noted by Zhai et al., 2022; Wang et al., 2021).
4 DISCUSSION
We introduced quasi-Monte Carlo latent variable models, or QLVMs simple but effective ap- proach to achieve very low-dimensional embeddings based on a deep generative model. While these models struggle to capture finescale details in complex datasets (e.g. high-resolution images), they are an attractive option for practitioners who prioritize interpretability over sample quality, We view QLVMs as strong alternative to popular methods like UMAP (McInnes et al., 2018), (SNE (Van Der Maaten, 2009), and ISOMAP (Tenenbaum al., 2000), which are popular for learning 2d or 3d embeddings, but do not learn an explicit data generating process.
Because they can be directly visualized, QLVM embeddings are amenable to a variety of transpar- ent manipulations and post hoc analyses such clustering ($3.3), calculating geodesic paths (ap- pendix C; Arvanitidis et al. 2018), and visualizing smoothness of the decoder mapping (Fig. 5D,H). One can easily imagine additional applications, such estimating mutual information or penalizing the Lipschitz constant of the decoder network during training (see e.g. Kinoshita et al., 2023). All of these analyses quickly become nontrivial as additional latent dimensions are considered.
4.1 OTHER RELATED WORK
We focused on standard VAEs and IWAEs as competitors to QLVMs. Although these are the most well-established models, there is broader literature this subject including work that explores quential and Markov Chain Monte Carlo methods (Salimans et al., 2015; Goyal et al., 2017: Huang al., 2018; Thin al., 2021), iterative refinements of the variational distribution output by the en- coder network (Kim et al., 2018; Marino et al., 201 8; Emami et al., 2021), and leveraging ensembles of models for improved importance sampling (Kviman al., 2022). In general. these methods come with nontrivial hyperparameter choices such as the number of MCMC steps, stochastic transition kernels, and annealing schedules. Moreover, they often incur additional computational costs for example, by fitting ensembles of encoder networks (in Kviman et al., 2022) by introducing and noteworthy that Thin et al.
optimizing backward Markov kernels (e.g. Salimans et al., 2015). (2021) motivate this general line of work by stating that IWAEs scale poorly to high-dimensional la- tent spaces, which is in direct contrast to our goal to identify very low-dimensional representations. In appendix F, we show direct comparison between QLVMs and iterative amortized inference Marino et al. (2018) on 2D MNIST embeddings, confirming that the latter is considerably more time intensive train and performs similarly the IWAE baseline shown in the results section.

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4.2 LIMITATIONS
Sample quality. Deep latent variable models are leveraged for a variety of applications in science and engineering. In certain settings, such as image or audio synthesis, the primary goal generate high-resolution, high-quality, and diverse samples. In these cases, state-of-the-ard methods make use of high- sigh-dimensional and hierarchically structured latent variables (Vahdat & Kautz, 2020; Rombach et al., 2022). As noted in section 2, the quasi-Mont Carlo approximation employed by QLVMs does not scale well to high-dimmnnsoonl spaces. Thus, our work has limited utility to practitioners who are willing to sacrifice interpretability for state-of tatte-of-the-rr the performance and sample quality. However, QLVMs are appealing in applications where interpretability is prioritized.
Scaling to high-diieensiona latent spaces. QLVM performance is driven by densely sampling the latent space with a discrete lattice of m points. Increasing m improves performance, but can be computationally costly. Fortunately, in low-dimensional spaces, these costs are not nearly bad as one might imagine (see Fig. 4). In principle. IWAE models with expressive and well-trained encoders could approximate the marginal log likelihood with smaller number of latent samples than a QLVM. However, this can be difficult to achieve in practice as seen both in previous work (Rainforth et al., 2018) and in results in Fig. 4. Furthermore, as discussed in section 2.3, IWAEs suffer several computational drawbacks that are overcome by QLVMs. The main computational limitation of QLVMs is their ability to scale high-dimensional latent spaces-due to the curse of dimensionality, the number of latent samples must increase exponentially with the latent dimension to maintain similar density of sample points.
Interpretability of oow-dimeniional latent spaces. While very low-dimensional embeddings have advantages, they must also be interpreted with care. For example, the 3dshapes dataset contains "ground truth" features. These features must somehow be mixed together when we fit a 2D QLVM. Indeed, we find that these six factors follow smooth nonlinear profiles over the embedding space, as shown in Fig. 6. The embedding is clearly organized with respect to three of the six features (wall hue, floor hue, and object hue) but the patterns with respect to the remaining three features are not easy visualize. Of course, it is impossible to linearly disentangle more than two features in a 2-dimensional latent space, so these limitations are not unique to QLVMs. Nonetheless, this is a broader limitation of visualization techniques that future research may wish to address.
The optimal embedding identified by deep latent variable models is generally not unique (Hyvärinen& & Pajunen, 1999) unless further assumptions are introduced (Khemakhem al., 2020). In their simplest form, QLVMs do not address this concern. Future work should investigate the extent to which this non-uniqueness impacts post-hoc analyses such clustering.
Identifiability.
4.3 POSSIBLE EXTENSIONS
One can imagine many refinements to the approach we've outlined in this paper. First, it would be interesting to explore whether some of the ideas discussed in $4.1 related to Monte Carlo might be incorporated to refine the QLVM objective. For example, one could use lattice to initialize an adaptive importance sampling routine (Bugallo et al., 2017). Such improvements could enable the model to utilize the latent space at finer scale than is specified by fixed lattice, thereby improving the quality of generated samples and addressing a key limitation.
Another extension would be to incorporate additional domain knowledge as conditional variables that modulate the latent space- a principle that is already well-established in the broader literature (Sohn al., 2015). In many applications, we expect this aid model interpretability as well as expressivity. For example, similar body movements can be executed at faster or slower speeds; this motivated Wu et al. (2025) to use conditional VAEs to disentangle speed from unsupervised latent dimensions (see also Costacurta et al., 2022). Similarly, in acoustic datasets (c.g. in Fig. 5E-H), it would make sense condition on features such as syllable duration, volume, and estimates of the fundamental frequency. We expect that providing the decoder with these statistics would also im- prove the quality of QLVM reconstructions, while preserving the interpretability of the embedding.
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A ADDITIONAL MATHEMATICAL DETAILS
A.1 LATTICES
Throughout, randomly shifted lattice defined modulo 1. That is, if u1..... um define d- dimensional lattice, we set 2 = U + A [u + AJ where A ~ Unif([O, 112) is random shift and |-] denotes the flooring operation.
A latrice is a set of points L C R d 2 that
xtye L, L), and 2. contains repeat points (Vx, y L, + y, 3e 0 such that || y c)
1. is closed under addition and subtraction (x,
A lattice rule is a lattice L constrained to the unit hypercube. In this work we use rank-1 lattice ib
b E Zd in particular, we use Korobov lattice rules, which are lattice rules such that u m rules, which = [1, a, a² - I......,, ad- lattice is special Fib(k) is the k-th element of the Fibonacci sequence. As Fibonacci lattice rules have been shown to optimally reduce integration error over the unit square, spaces; we use more general Korobov lattice rules for three-dimensional latent spaces. See Dick al. (2022); Owen (2023) for further details regarding lattice rules.
a4 Il. a a (2.3..... m 1}. The Fibonacci m
of Korobov lattices in which dl = = Fib(k - 1),
Fib(k), where
those for all two-dimensional latent
A.2 MEAN-SHIFT CLUSTERING
The mean-shift algorithm is mode clustering method defines clusters based on modes of density of the data. This algorithm begins with a of seed points and finds modes by iteratively moving these seed points towards areas of high density in local neighborhoods, defined by bandwidth parameter > 0. Starting from (0)), updates are defined as:
Eiz;K
1
(1+1)
(8)
E, K
where K-)) is smoothing kernel; in our experiments we set K(u) = exp(-u²). Given that our QLVM gives density estimate, we can replace points 2; with our lattice points, ž,, and density estimates with our aggregate density Ex [p(2 x)]. In neighborhood N₁, around our current mode estimate:
(++ = EiEN, 2.E.Ip(2.1=))
Σi,eN, E. =x
Upon convergence of all initial seeds to local modes, seeds within the set bandwidth of each other are merged, favoring seeds with the highest overall density. See (Wasserman, 2018) for further details.
B EXPERIMENTAL DETAILS
All models were trained to convergence using Adam with learning rate=0.001, 81 0.9, Ag 0.999. MNIST and gerbil vocalizations were trained using Bernoulli log probability over pixels, all other datasets IWAEs, variational posteriors were Gaussian distributions. latent priors were standard Gaussian distributions. QLVM training took between 30 minutes (MNIST) and 9 hours (3dShapes) on one NVIDIA H100 GPU. For all IWAEs we used m 10 importance samples.
trained using Gaussian log probability with variance= 0.1. For VAEs and
B. DATASETS
B.1.1 MNIST, Celeb-A, AND 3dShapes
For both MNIST (LeCun et al., 2010) and Celeb-A (Liu et al., 2015) we used default train/test splits. Celeb-A images were downsized to 80x80 pixels and converted to grayscale. For 3dShapes (Burgess
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0.80
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0.96
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0.00
2.00
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12.86
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Figure Al: Example samples from the 3dShapes dataset. Numbers indicate latent factor values. From top to bottom: Floor hue, wall hue, object hue, scale, object shape, orientation.
& Kim, 2018), models were trained using an 80/20 train/test split. See Fig. Al for examples of the 3dShapes data.
1.1.2 ZEBRA FINCHI AND GERBIL VOCALIZATIONS
We used vocalizations from single adult zebra finch, Blu285 (Goffinet et al., 2021), and vocaliza- tions from three different families of gerbils (Peterson et al., 2024). Vocalizations were segmented and processed into spectrograms using Autoencoded Vocal Analysis (Goffinet al., 2021). Models on both datasets were trained using an 80/20 train/test split.
B.1.3 CMU MOTION CAPTURE
We used motion capture data from the Carnegie Mellon University Motion Capture dataset (https: /mocap.. cs edu/) specifically, subject 54 (human pantomiming animal behaviors). Each trial consists of a sequence of poses; for each of the 44 non-torso joints in the pose we extract the sine and cosine its Euler angle. We perform the same transformation and from these compute global (torso) translational velocities. Each data point fed to the model was a stack of 10 contiguous transformed frames; models were trained using an 80/20 train/test split.
global (torso) pose angles,
B.2 ARCHITECTURES
See Fig. A2 for all decoder architectures used for all experiments. VAE and IWAE encoders had the same structure as decoders in the reverse order, with ReLU nonlinearities replaced by hyperbolic tangent nonlinearities for stability during training.
B.3 PERFORMANCE BY NUMBER OF SAMPLES QLVMS AND IWAES
To compare the computational cost (vs. model performance) in IWAEs and QLVMs, we trained each model on MNIST, Celeb-A, and the zebra finch dataset, varying the number of importance samples k in IWAEs and number of lattice points m for QLVMs. For IWAEs, we used:
MNIST, E [5, 10, 15, 20, 25, 30, 35, 40, 85, 140, 225]
Celeb-A, à [5, 10, 15, 20]
zebra finch, & [5, 10, 15, 30, 60, 70,
For QLVMs, we used:
MNIST, m E [55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368) Celeb-A, m E [55, 89, 144, 233, 377, 377,377,700000 610,987) 987]
zebra finch, [55,88 89, 144, 233, 377, 610, 987]
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A
B
Upsample cell
MNIST
Celeb-A
Gerbil
Zebra finch
Linear (d,500)
Linear (d,2048)
Linear (d,64)
Linear (d,2048)
Layer Norm
Linear (500,784)
Linear (2048,1600)
Linear (64,256)
Linear (2048,4096)
TranTrannsose (3x3)
Sigmaid
Res Cell
Linear (256,1024)
4x ConvTranspose (3x3)
Layer Norm
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Figure A2: (A) Neural network cells used in models. (B) Model decoders used in all experiments. All linear and convolutional layers prior to the sigmoid output were followed by a ReLU nonlincarity.
Architectures were the same as those used in all other experiments; we trained three models for each value of m and k.
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C LATENT TRAVERSALS AND GEODESICS
Because of the low dimensionality of our latent space, we can perform linear latent traversals (as typically carried out to explore dimensions of LVMs) or find geodesics based on the aggregate density over the data. Additionally, since our latent space is constrained to the unit d-cube, we can traverse over all values of each latent variable. We see that, for the most part, moving smoothly through latent space results in smooth changes to the output images. Abrupt changes occur what appear be class boundaries.
in Fig. A3 for each dataset. each, we
We can probe these regions of what seem to be abrupt change based either on reconstructions or on the aggregate posterior. We demonstrate one such case on the gerbil vocal dataset in Fig. A4. Traveling parallel to what appears to region of low density results in little change in the output, while there is an abrupt change the output spectrogram the moment that boundary is crossed.
Finally, we can compute geodesics connecting points in the latent space (Arvanitidis et al., 2018). Specifically, we construct weighted graph over this lattice by considering any point to be connected to its close neighbors on the lattice, with weight determined by the aggregate posterior density ratio of that point to its neighbor. This assigns a high cost to moving from point with high density to point with very low density, while prioritizing moving towards points with highest density. We then use Dijkstra's is algorithm to find the shortest path on this graph between two query points. The output of this procedure is visualized in Fig. A5, along with reconstructions from the geodesic.
9 4
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Figure A3: Traversals over the entire latent space for all datasets. Due to the constraints on our latent space (periodicity on the unit d-cube, 2-dimensional) we are able fully explore the latent space for each model.
NNote that the density ratio defines one choice of a local metric, but others are also possible such as the pull back metric defined by the Fisher information matrix (Arvanitidis et al., 2018). We plan to pursue and compare these possibilities in future work.
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Figure A4: Latent traversal for the gerbil vocalization dataset. The top row is traversal roughly parallel to what seems to be a class boundary; the bottom row crosses this boundary. The red dot and outlined red spectrogram correspond to the point which this traversal crosses this boundary. This crossing accompanied by a sharp change in the reconstruction.
Figure A5; Geodesic from centroid to centroid 15 (in Fig. 5B), and corresponding reconstructions
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D 3dShapes POSTERIORS AND JACOBIANS
Here, we visualize the aggregated posterior and squared Frobenius norm of the Jacobian for the 3dShapes dataset analyzed $3.4. The motivation for these analyses are described in $3.3, where they are demonstrated on MNIST and a dataset of gerbil vocalizations. Unlike in those datasets, we see no evidence of clustering- -ie. no peaks in the aggregated posterior or ridges in the scale of the Jacobian in the QLVM embedding of the 3dShapes dataset. Instead, we see roughly uniform density over the entire latent space, reflecting the continuous nature of the generating latent factors for this dataset (Fig. A6).
Aggregated posterior
Smoothed f(>2)()))
Latent dimension
Latent dimension
Latent dimension
Figure A6: Aggregated posterior (left) and smoothed Frobenius norm of the decoder Jacobian (right). Unlike in MNIST and the gerbil vocal dataset, there is visible clustering.
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E MODEL ALTERATIONS
E.I MC,QMC,AND RQMC TRAINING
The bound of eq. (5) holds for any set of Monte Carlo samples how important the RQMC sampling scheme that we used? We trained networks with the same architecture and same number of latent samples per data point on MNIST using uniform MC samples and a fixed QMC lattice, finding that the RQMC sampling scheme that we used in the main text yielded small, but consistent improvements in performance. We hypothesize that this is because of the structure of lattice combined with the uniform random shift; the uniform tiling of the latent space forces the model to explore the entire latent space every batch, while the shift ensures that the gaps in the latent space left by the lattice are explored across batches.
-118.8
Bound on MLL
-119.2
-119.6
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MC
QMC
ROMC
Figure A7: Comparison between model performance on MNIST test set using MC samples, a fixed QMC lattice, RQMC (as in main text) for training. 10 models were trained per condition, RQMC consistently displays best performance.
E.2 ALTERNATIVE LATENT SPACES
While lattices perform best when used for integration over periodic functions (see Owen, 2023, Chapter 16, section 7) it may be the case that flexible neural network decoder does not require periodicity (or can learn it implicitly) in order learn good representation for data. We demon- strate in Fig. A8 that this is not the case; rather, it impairs model performance (in terms of log evidence). As such, appears that both enforcing periodicity helpful for learning a good decoder.
Alternatively, rather than using flat torus basis, we could use a Gaussian inverse CDF basis to effectively have Gaussian prior over the latent space, while still relatively constraining the domain of the model. In Fig. A9, we compare performance of this basis to the QLVMs trained throughout the rest of the paper. This version performs similarly (but slightly worse) than QLVMs with uniform prior and periodic boundary conditions on the latent space. We chalk up this minor difference to the fact that the Fibonacci lattice rule is not optimized for the Gaussian case. Regardless, the QLVM with Gaussian prior outperforms VAE and IWAE models with the same prior.
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I
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Non-periodic uniform QLVM
Uniform QLVM
Figure A8: Comparison between model performance on MNIST using a non-periodic decoder, or a standard QLVM (as in the main text). 10 models were trained per condition. The standard QLVM consistently displays best performance.
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Figure A9: (A) Comparison between QLVMs trained using the default uniform prior, Gaussian prior, 2D IWAEs, and VAEs. each case, both QLVMs outperform IWAEs and VAEs, with Gaussian QLVMs slightly underperforming uniform QLVMs. (B) Latent embeddings of MNIST the uniform QLVM and Gaussian QLVM.
F COMPARISON TO ITERATIVE AMORTIZED INFERENCE
An alternative approach to improving the performance of VAΣs is to iteratively refine the varia- tional distribution output by the encoder network, as proposed by Marino et al. (2018). While this strategy was shown to be effective for the higher-dimennsonall models, demonstrate that it still underperforms 2D QLVMs (Fig. A10).
MNIST
Negative bound on MLL
QLVM
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Figure A 10: Performance vs. computational cost curves for 2D QLVMs (red). VAEs (green), IWAEs (purple), and iterative amortized inference-trained VAEs (orange) on MNIST. Curves closer to the lower left quadrant of each plot indicate more favorable tradeoff.
G COMPARISON TO HIGHER-DIMENSIONAL VAES
We additionally compared 2D QLVMs to VAES and IWAEs of higher dimensions. Higher- dimensional IWAEs and VAEs outperformed 2D QLVMs quantitatively (Fig. A11); however, QLVMs qualitatively had similar reconstructions to up to 8D VAEs (Fig. A12). On simpler datasets, the improvements of higher-dimensional VAEs relatively minor. However, samples from the QLVM prior both show higher sample quality and diversity than low-dimensional VAEs, and their samples more accurately reflect data than nigher-dimensione VAEs (Fig. A13).
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MNIST
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IWAE VAE IWAE VÁE IWAE VAE IWAE
(d=8) (d=16) (d=16) (d=32) (d=32) (d=64) (2=))
Figure All: Bounds on marginal log likelihood on heldout test data for QLVMs, VAEs, and IWAEs of the same and larger dimensions; orange bars 2D, gray bars are higher dimensional. Error bars indicate standard deviation across five random seeds.
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Figure A12: Reconstructions from models from Fig. A11; 2D QLVM reconstructions are compara- ble with up to 8D VAES.
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Figure A13: Samples from the prior for QLVMs and VAEs from Fig. A11; samples from QLVM prior resemble data more closely than samples from VAE priors.
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Reviews:
[{"review_id": "FWnXqIygn2", "reviewer": "Reviewer_2qje", "summary": "The authors propose a latent variable model that directly optimizes the marginal likelihood using Monte Carlo integration, without using an encoder-based approximate posterior. For uniform priors, the authors provide appropriate integration schemes when the latent space has small dimensionality. The experimental results show that the proposed approach is competitive with related deep generative models, and the learned latent representations appear meaningful.", "questions": "Q1. The structure of the latent space is somewhat unclear due to the periodic boundary. Could the authors elaborate on the choice $\\mathbf{z} = (\\sin \\mathbf{z}, \\cos \\mathbf{z})$ and its implications? What is the topological structure of the resulting latent space?\n\nQ2. In Fig. 1, it appears that blue points are always associated with batch 1 and red points with batch 2. Are the batches fixed throughout training, or do they change? In other words, do data points in batch 1 always interact with the same set of latent points?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The proposed method is conceptually simple, straightforward to implement, and performs well in the conducted experiments.\n- The integration schemes, together with the uniform prior, appear to leverage the latent space effectively.\n- The paper is generally well written and easy to follow.\n- I think that the method provides a meaningful alternative to UMAP-style approaches, with the advantage that it includes a decoder.", "weaknesses": "- As acknowledged by the authors, scalability is a clear limitation of the approach.\n- While the experimental results are promising, it is not completely clear to what extent this approach opens new research questions beyond the presented scenarios.", "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": "Z6VwpnIwhB", "reviewer": "Reviewer_6SFt", "summary": "This paper investigates the performance of QMC in computing the marginal log likelihood in very low-dimensional settings. This resulted in a model that, although computationally demanding, did not require an encoder and outperformed VAEs and IWAEs on simple datasets, such as MNIST. Given the dimensionality limitation of such models, the authors propose this method as an alternative to UMAP or t-SNE. In opposition to these algorithms, QLVMs benefit from additional capabilities, such as latent traversal or density estimation, which are essential for unsupervised data analysis, ensuring transparent results.", "questions": "- In 3.1, VAEs in very low-dimensional spaces may suffer from posterior collapse [1-3], it could easily be checked (e.g., following the method proposed in [3] to monitor passive variables). Such results would provide a good explanation for the poor performances of VAEs in this setting. I am not sure whether IWAE can also suffer from posterior collapse, but this could be worth exploring.\n- Also in 3.1, VAEs tend to behave in a polarised regime [2-4] where some latent variables are used to decrease the KLD while others encode meaningful information but depart from the prior. Could what is observed in Fig. 3 be a consequence of this?\n- In Fig. 3 B, which line corresponds to which model?\n- Have the authors tried QLVM on high-dimensional tabular data (e.g., gene expression)? Which type of results would they expect?\n- l. 123 it should be \")]\" instead of \"])\"\n_________\n\n**References**\n\n[1] Dai, B., Wang, Z., & Wipf, D. (2020, November). The usual suspects? Reassessing blame for VAE posterior collapse. In International conference on machine learning (pp. 2313-2322). PMLR.\n\n[2] Rolinek, M., Zietlow, D., & Martius, G. (2019). Variational autoencoders pursue PCA directions (by accident). In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (pp. 12406-12415).\n\n[3] Bonheme, L., & Grzes, M. (2023). Be more active! understanding the differences between mean and sampled representations of variational autoencoders. Journal of Machine Learning Research, 24(324), 1-30.\n\n[4] Dai, B., Wang, Y., Aston, J., Hua, G., & Wipf, D. (2018). Connections with robust PCA and the role of emergent sparsity in variational autoencoder models. Journal of Machine Learning Research, 19(41), 1-42.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "This paper investigates a simple yet overlooked way of computing the marginal likelihood and proposes a creative approach to use this in the restricted setting of 3 or fewer latent dimensions. I believe that this paper can have a strong impact on the unsupervised learning community, especially for people doing exploratory data analysis on high-dimensional data. The paper was easy to follow, and I really enjoyed reading it. Well done to the authors, great work!", "weaknesses": "- The posterior collapse issue of VAEs when reducing the number of latent dimensions too much has been overlooked. It would be great to add this to the discussion in 3.1.\n- Fig. 3 B do not explain which models are associated with each line.", "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": "PloSQo34Rz", "reviewer": "Reviewer_Q9q7", "summary": "This paper takes an interesting approach to latent variable modelling. The presentation contrasts the approximations done using amortized inference with a more direct inference by using a monto carlo estimate of the marginal log-likelihood. The approximation to the marginal log-likelihood is done using uniform prior which means that the latent space needs to be bounded and for computational reasons very low-dimensional in order to get a good estimate. The paper discusses the pros and cons of using this more \"natural\" approach compared to an amortized inference mechanism used in VAEs.", "questions": "Given that you are able to compute the Jacobian, can't you then represent the pull back metric in the latent space? This would allow you to compute more principled Geodesics compared to the approach you are now taking.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This is well written paper that is very easy to understand. In some sense I think the narrative of explaining this and continously relating this to VAEs is a little bit too specific, in some ways this is a much more natural approach to the problem of a latent variable models compared to an amortized inference scheme. That is just my personal oppinion and not something that has been reflected in the score. The flip side of this is that the latent space is required to be very low-dimensional in order to be able to compute the approximation. As such, and this the authors do very well, it shouldn't really be put in the same context as VAEs or other generative models but rather UMAP t-SNE and possible a GP-LVM as its where these models are useful this approach will be applicable. Differently from the first two though, this is a generative model and allows for the latent location to be infererred for new data, due to the uniform latent prior, which is more challenging with spectral dimensionality reduction methods.\n\nI liked the discussion around the failure modes of the amortized approximate posterior. I think your intuitions about this makes sense and it would be nice to evaluate this a bit more to get a more. There is a very nice paper [1] that discusses the interplay of the complexity of the back and the generative mapping in an auto-encoder setting. There might be some information that is relevant there for future work.\n\n[1] Barber, D. (2014). Implicit Representation Networks.", "weaknesses": "While I think the paper is a nice study and tells a story about how one can approach latent variable modelling in a more direct way there is also a question about the limitations of the paper. The first thing I think would have been interesting to see is the implication of the uniform prior over the latent represenation and what effects it does have. It would be great to have a discussion on more informative priors and how the approach would change in that case and the impact on the results. I don't fully understand the results right at the end of the Appendix which I assume are supposed to relate to this.\n\nThe authors state and are very honest with the fact that the approach is very computationally expensive. It would be nice to see what this actually means in-terms of numbers and how many samples \\(M\\) that is used in the experiments and the dependency of this.", "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": "TUBnNiPRos", "reviewer": "Reviewer_YdPh", "summary": "This paper investigates using explicit integration over the latent variable in deep latent variable models as an alternative to the amortized variational inference approach of VAEs. As this integral is intractable the authors propose using \"quasi monte-carlo\" integration; rathering than estimating via samples from the prior, the authors sample a randomly shifted lattice of points in the latent space to approximate the integral. The authors argue that this approach achieves considerably better results than variational auto encoders in cases where the latent space is low-dimensional enough that this approach is computationally feasible, i.e. 2 and 3 dimensions, cases where the latent embeddings may be used for visualization purposes. In their experiments, the authors show better performance compared to VAEs and importance-weighted auto encoders as measured by test loss and reconstruction. They also compare the approach to other low-dimensional embedding approaches tailored to visualization, such as UMAP.", "questions": "- Is the variational posterior approximation for the VAE and IWAE baselines still Gaussian in your experiments? \n  - If so, how do you enforce the boundary constraints for the uniform prior and estimate the KL-divergence etc? \n  - Does this affect the variance of the training objective e.g. compared to a Gaussian VAE where the KL-divergence can be computed in closed form?\n- How is the test loss in figure 2 computed? \n  - All three methods are computing different lower bounds on the MLL, so my question is: is this comparing learned decoder performance by evaluating all of the learned models with the same lower bound?\n  - Or alternatively is it reporting the same loss used to train each model?", "limitations": "", "rating": 2, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "**Writing**\n- The paper is clear and well written with a nice description of the motivation of the problem. \n- The paper includes a good discussion of related work and identifies many relevant papers from recent literature\n\n**Methodology**\n- Using randomly shifted lattices to tile the latent space for training is a clever idea and there does seem to be some notable performance improvements in some cases.\n- There is a good exploration of the value of the lattice based approach in the appendix\n\n**Results**\n- The paper includes some excellect visualizations that make a compelling case for a deep-latent variable model with 2-dimensional uniform prior as a method for generating useful visualizations for high dimensional datasets.\n- The authors used an interesting variety of datasets to showcase the performance of the approach", "weaknesses": "**Novelty**\n- As the authors acknowledge, this idea is not inherently novel, though it has generally been dismissed as impractical in the past. I think that showing a compelling use case for this approach could be a worthwhile contribution, but it sets a high bar for the experimental results.\n\n**Computational costs**\n- This method has a high computational cost, even for the 2-d and 3-d cases discussed. The authors only test with relatively simple datasets and networks.\n- The authors don't provide any quantitative comparison of the computational costs of different methods. This is a notable omission as computational concerns are the primary reason this approach hasn't been explored before.\n\n**Framing**\n- The authors frame this approach as generally better for low-dimensional latent variable models, but notably change the prior from the standard Gaussian used in most previous work to a uniform, without comparison their approach using a Gaussian prior. \n- I think the uniform is a valid choice in this context for something like visualization, but it should be clear if this is not necessarily a general-purpose substitute for VAE-style methods.\n\n**Baseline comparison**\n- I'm not convinced that the comparison to baselines is fair.\n- Looking at figure 2, the VAE seems to be suffering from posterior collapse. In my experience it's definitely possible to train a VAE for MNIST with 2-latent dimensions that performs better than what is shown.\n- The IWAE baseline only used 10 samples, which seems quite small given the comparison to QLVMs. I would prefer to see a computation-time matched comparison to better understand if the QLVM approach is actually superior.\n- The authors dismiss improved variational methods such as iterative amortized inference based on their computational expense and hyper parameters, but I would see these approaches as an important baseline. If they are truly less practical than this approach, I would like to see that justified experimentally.\n\n**Experimental details**\n- There are several details of the experiments that are not properly covered in the text as mentioned below.", "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)