Laplacian Autoencoders for Learning Stochastic Representations
Authors: Marco Miani
Frederik Rahbæk Warburg
Pablo Moreno-Muñoz
Nicki Skafte Detlefsen
Søren Hauberg
Venue: NeurIPS 2022 Accept (2022)
Keywords: Laplace Approximation
Bayesian Learning
Unsupervised Representation Learning

Abstract: Established methods for unsupervised representation learning such as variational autoencoders produce none or poorly calibrated uncertainty estimates making it difficult to evaluate if learned representations are stable and reliable. In this work, we present a Bayesian autoencoder for unsupervised representation learning, which is trained using a novel variational lower-bound of the autoencoder evidence. This is maximized using Monte Carlo EM with a variational distribution that takes the shape of a Laplace approximation. We develop a new Hessian approximation that scales linearly with data size allowing us to model high-dimensional data. Empirically, we show that our Laplacian autoencoder estimates well-calibrated uncertainties in both latent and output space. We demonstrate that this results in improved performance across a multitude of downstream tasks.

Laplacian Autoencoders for Learning Stochastic Representations
Marco Miani', Frederik Warburg', Pablo Moreno-Muñoz, Nicke Skafte Detlefsen, Søren Hauberg {mmia, frwa, pabmo, nsde, sohau)@dtu.dk Technical University of Denmark
Abstract
Established methods for unsupervised representation learning such variational autoencoders produce none or poorly calibrated uncertainty estimates making it difficult evaluate if learned representations are stable and reliable. In this work, we present a Bayesian autoencoder for unsupervised representation learning, which is trained using a novel variational lower bound of the autoencoder evidence. This is maximized using Monte Carlo EM with a variational distribution that takes the shape of Laplace approximation. We develop new Hessian approximation that scales linearly with data size allowing to model digh-dimensional data. Empirically, we show that our Laplacian autoencoder estimates uncertainties in both latent and output space. We demonstrate that this results in improved performance across a multitude of downstream tasks.
1 Introduction
Unsupervised representation learning is brittle matter. Consider the classic autoencoder (AE) (Rumelhart et al., 1986), which compresses data æ to a low-dimensional representation # from which data is approximately reconstructed. The nonlincarity of the model implies that sometimes small changes to data æ give a large change in the latent representation z (and sometimes not). Likewise, for some data, reconstructions the model does not have built-in quantification its uncertainty, and we cannot easily answer when the representation is reliable and accurately reflects data.
of low quality, while for others it is near perfect. Unfortunately,
The celebrated variational autoencoder (VAE) (Kingma and Welling, 2014; Rezende et al., 2014) address this concern directly through an explicit likelihood model p(=]zz) z) and a variational approxi- mation the representation posterior p( = x). Both these distributions have parameters predicted by neural networks that act similarly the encoder-decoder pair of the classic autoencoder.
But the VAE'S quantification of reliability reliable? To investigate, we fit dimensional latent representation to the MNIST dataset (Lecun et al., 1998), and illustrate the predicted uncertainty of p(x]2) 2) in Fig. The model learns to assign high uncertainty to low-level image features such as edges but predicts its smallest values far away from the data distribution. Not only such behavior counter-intuitive but also suboptimal terms of data likelihood (Sec. 1.1). Retrospectively, this should not be surprising the uncertainty levels away from the data are governed by the extrapolatory behavior the neural network determining p(=[=z). This suggests that perhaps uncertainty should be derived quantity rather than predicted one.
with a two-
Denotes equal contribution; author order determined by simulated coin toss.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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[a] VAE
(b) LAE (post-hoe)
LAE (ouline)
Figure 1: 2D latent representation of MNIST overlaid heatmap that describes the decoder uncertainty (yellow/blue indicates a low/high variance of the reconstructions). To the right of the latent spaces, show the mean and variance of reconstructed image (yellow indicates high values). (a) The VAE learns to estimate high variance for low-level image features such as edges but fails at extrapolating uncertainties away from training data. (b) Applying post-hoc Laplace to the AE setup shows much better extrapolating capabilities, but fails in estimating calibrated uncertainties in output space. (c) Our online, sampling-basee optimization of Laplacian autoencoder (LAE) gives well-behaved uncertainties in both latent and output space.
From a Bayesian perspective, the natural solution is to form an (approximate) posterior over the weights of the neural networks. To investigate, we adapt a state-of-the-an implementation of post-hoc Laplace approximation (Daxherger et al., 2021) of the weight posterior to the autoencoder domain. This amounts training a regular autoencoder, and thereafter approximating the weight uncertainty with the Hessian of the loss (Sec. 1.1). Fig. lb shows that uncertainty now grows, as intuitively expected, with the distance to the data distribution, but there seems to be little semantic structure in the uncertainty in output space. This suggests that while the post-hoc procedure is computationally attractive too simplistic.
In this paper we introduce a new framework for Bayesian autoencoders in unsupervised represen- tation learning. Our method takes inspiration from the Laplace approximation to build variational distribution on the neural network weights. We first propose a post-hoc T.A for autoencoders; show- casing good out-o--distribution detection capabilities, but lack of properly calibrated uncertainties in-distribution. To address this, we develop fast and memory-efficient Hessian approximation, which allows us to maximize variational lower bound using Monte Carlo EM, such that model uncertainty is key part of model training rather than estimated post-hoc, Fig. Ic gives an example of the corresponding uncertainty, which exhibits a natural and semantically meaningful behavior.
1.1 Background
The VAE is a latent variable model that parametrize the data density p(x) f p(x: (2)(2)((( using a latent variable (representation) z. Here p(z) is a, usually standard normal, prior over the representation, and p(z|2) is likelihood with parameters predicted by neural network.
The nonlinearity of the likelihood parameters renders the marginalization of intractable, and variational lower bound of p(æ=) is considered instead. To arrive at this, one first introduces variational approximation q(2) a>) Ri p(2(ær) and write p(xx) = (gy()))) [p(x] (/(]//))/)) A lower bound on p(x:) then follows by a direct application of Jensen's 's inequality,
logp(x) 2 L Eng(+()) [logp(r|=)] (I<((()))))l<()))
(((2),, o( (=)), we see that optimally (2)) should be as large as possible away from training data in order to increase p(x) on the training data (Appendix F). Yet this is not the observed empirical behavior in Fig. la. Since the o² network is left untrained away from training data, its predictions depend on extrapolation. In practice, o² takes fairly small values near training data (assuming the mean μ provides reasonable data fit), and o² extrapolates arbitrary terms of data likelihood. Similar remarks hold for other likelihood
If we momentarily assume that p(x/ ) Na(
even if this suboptimal models p(x] (2) and encoder distributions (<]|=): relying neural network extrapolation predict uncertainty does not work.
The Laplace approximation (Laplace, 1774; MacKay, 1992) is an integral part of our proposed solution. the context of Bayesian neural networks, we seek the weight-posterior D)) x exp(-L(D;;)) 0)), where 0 are network weights, D is the training data, and L is the applied loss

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Pp
xx
Z
S(p
Model overview. We learn a distribution over parameters such that we can sample encoders E and decoders dæ. This allow us to compute the empirical mean and variance in both the latent space & and the output space (x.)) ) = (e((e((x.))))
Figure 2:
function interpreted as unnormalized log-posterior. This generally intractable and Laplace's approximation (LA) amounts to second-order Taylor expansion around a chosen weight vector 6*
log (0 = - ( V' + + (0 T 0*) + O(1 11))
where we use the short-hand L = L(D; The approximation, thus, assumes that p(( is Gaussian. Note that when 0* is term is negative semi-definite, We provide more details on the Laplace approximation in Appendix B. In practice, computing the full Hessian is too taxing both in terms of computation and memory, and various approximations are applied (Sec. 3).
estimate, the first order term vanishes and the second order
2 Laplacian Autoencoders
We consider unsupervised representation learning from i.i.d. data { consisting of observations Hn R. We also define a continuous latent space such that representations nn E RR R Similar to AES (Hinton and Salakhutdinov, 2006), we consider two neural networks e -R and de: R -R widely known as the encoder and decoder. These have parameters {t,, Φ). We refer the composition of encoder and decoder fe d. cs.
Model overview. The autoencoder network structure implies that we model the data being distributed on K dimennsional manifold parametrized by 0. We then seek the distribution of the re- construction Bree fa( given observation æ, where the uncertainty comes from being unknown,
p (res x. f) f) 0, f)].
Notice that explicitly condition on f. which is the operator 0 - fe- even if this is not stochastie; this conditioning will become important later on to distinguish between the distribution deduced by f and its linearization f(t), Mimicking the standard autoencoder reconstruction loss, we set p(Zrce 0,, x, f) = (2 1). Since p(O ax, f) is unknown, the reconstruction likelihood Eq. (3) intractable, and approximations are in order. Similar to Blundell et al. (2015), we resort to a Gaussian approximation, but rather than learning the variance variationally, opt for LA. This will allow us to sample NNS and deduce uncertainties in both latent and output space as illustrated in Fig. 2.
Intractable joint distribution. Any meaningful approximate posterior over 0 should be similar to the marginal of the joint distribution p(0, Freee [x. f). This marginal
00 xr, Ex fm. (p( Fress x, f)]
which can be bounded log-scale using Jensen's inequality,
log p( 0
> Lo Em p(t () [log (ree x, f)].
(5)
Our first approximation LA of p(0)zz x. f) (0)xx, = _, where we postpone the details how to acquire 0, and H,. These will eventually be iteratively computed from the lower bound Eq. (5); hence the f index. Fig. 3 (a) illustrates the situation thus far: p(0 æ. f) is approximately Gaussian, but the non-linearity of ʃ gives p(aree az. J' a non-trivial density.
Linearization for gradient updates. Standard gradient-basse based learning can be viewed as lineariza- tion of f in 0, i.c. for a loss L l(fe), the gradient is VoL Jefe V v,l(f), where Jefe is the
3

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Xxec
Xyec
plo.. De. f)
K00 No
30Kaal a..x.
0
(a) Non linear f
(b) Linear f(t)
Figure 3: Illustrative example for fixed æ. The likelihood for fixed 9,. shown by the columns, are assumed Gaussian po x f) (x). We model the marginalised density p(0)æ, f) (first axis) over parameters with Gaussians. With the additional assumption of linear then p( x., f') (second axis) is Gaussian. This makes the joint distribution tractable.
Jacobian of f. In similar spirit, we linearize f in 0 in order to arrive at tractable approximation of p( x f). Specifically, perform Taylor expansion around 0,
fe(æ:) fo. (a) Jafe. (x) 0.) +0 (10 0.1l3)
) (z)
where (()) denote the associated first-order approximation of Under this approximation, the joint distribution p(0, Frea f) becomes Gaussian, p(0, (x, f(0) ze N (0, Tree (re E,), with
N4 (50))
0: and E, = J (o) (x) (J)fe(()) H. (x)) +1)-
H,¹
Jole. (x)
(t) (a))
This approximation is illustrated in Fig. 3(b). We provide the proof in Appendix D.4.
Iterative learning. With these approximations we can readily develop an iterative learning scheme which updates g xr, f).. The mean of this approximate posterior can be updated according to standard variational gradient step, 0.11 0. \VoL where
La 10 f) [log p x. (())),
(8)
a lower bound on Eq. (3), which we evaluate with a single Monte Carlo sample. Following the LA, the covariance of g (ae f), should be the inverse of the Hessian of log (x, f' at O,41. Since this is intractable, we instead compute the Hessian the lower bound Eq. (5)
1++1
V2 0.41 = (m.
V log log p( (0, ar (()) V log (0 x. f) 0::1
The last term can be approximated since V log q' To efficiently cope with the O-term, we introduce a parameter a, such that the final approximation is
(0 (0) f)le 0011 = - H, + O (110+1 - 0r1)).
H 2 (1 a:) H, + E Eppeaaee.)) v logg
@ree 0 xc (
(10)
0.+1
(1 H. Jo (2)
2 logp (Trec 0.+1, æx, 1(0) Jo
(11)
where (11) f is independent of which 0 we evaluate in and V2 log (rrec 0 æ, (()) ( is trivial to compute for common losses, i.e. for MSE the identity. The parameter a can be viewed as geometric running average that is useful for smoothing out results computed on minibatch instead on the full training set, similar to momentum- like training procedures. further allows for non-monotonically-icceeaa precision. Note that we revisit data during training, and the precision

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14=1
U111 02 AEB..
Vallgg (eree
20. fi]
Hoys (1 ajH, Jafo" V2.. log pia 10,x0 x 101%
Figure 4: Iterative training procedure. Given distribution q' over parameters, and a linearized function f(t), compute first and second-order derivatives to update the distribution parameters.
matrix is updated for every revisit. Thus, the forgetting induced by wanted behavior avoid infinite precision in the limit of infinite training time. In practice, we set a equal to the learning rate, where the practical intuition is that when is small, the network uncertainty decreases faster.
The overall training procedure is summarized in Fig. 4. We initialize gd as a Gaussian with 0, = and Hg I. We provide more details on the model, the linearization, and iterative learning in Appendix D.
Why not just...? The proposed training procedure may at first appear non-trivial, and it is reasonable wonder if existing methods could be applied similar results. Variational inference often achieves similar results to Laplace approximations, so could we use 'Bayes by Backprop' (Blundell et al., 2015) to get an alternative Gaussian approximate posterior over 0? Similar to the supervised experiences of Jospin et al. (2020), we, unfortunately, found this approach too brittle to allow for practical model fitting. But then perhaps a post-hoc LA as proposed by Daxberger et al. (2021) for supervised learning? Empirically, where the Hessian provides useful uncertainty estimates. Our online training moves in this direction the Hessian is part of the procedure, but this is not true for the post-hoc LA.
found it to be important to center the approximate posterior around point
Conceptually, we argue that our approach, while novel, not entirely separate from existing methods. Our reliance on lower bounds makes the method an instance of variational inference (Jordan et al., 1999; Opper and Archambeau, 2009b), and we maximize the bounds using Monte Carlo EM (Cappé, 2009). We rely a LA our choice of variational distribution, which has also been explored by Park et al. (2019). Finally, we note that our linearization trick Eq. (6) has great similarities to classic extended Kalman filtering (Gelb et al., 1974).
3 Scaling the Hessian to Large Images
The largest obstacle apply LA in practice stems from the Hessian matrix. This matrix has quadratic memory complexity in the number of network parameters, which very quickly exceeds the capabilities of available hardware. To counter this issue, several approximations have been proposed (Ritter et al., 2018; Botev et al., 2017; Martens and Grosse, 201.5b that improve the scaling w.r.t. to the number of parameters. The currently most efficient Hessian implementations (Dangel et al., 2020; Daxberger et al., 2021) builds on the generalized Gauss-Newton (GGN) approximation of the Hessian
V [())(x)) ~ Jot fe (x) V2 L (x Joa fo (z)
(12)
for a single layer 1, which neglects second order derivatives of w.r.t. the parameters. Besides, the computational benefits of this approximations, previous works LA (Daxberger et al., 2021) relies on GGN to ensure that the Hessian is always semi-negative definite. In contrast, the model presented in Sec. implies that GGN no longer practical and unprincipled trick, but rather the exact Hessian (Immer et al., 2021b).
Albeit relying on first order derivates, the layer-block-diagonal GGN, which assumes that layers independent of each other, scales quadratically with the output dimension of the considered neural network f- This lack of scaling is particularly detrimental for convolutional layers these have low parameter counts, but potentially very high output dimensions.
Expanding fo(r) with the chain rule, one realizes that the Jacobian can be computed as function of the Jacobian of the next layer. Fig. illustrate that an intermediate quantity M, which ai
initialised as 22 C(xree) can be efficiently backpropagated through multiplication with the Jacobian
5

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APPROXIMATIONS
MEMORY
TIME
Table Memory time complexity of Hessian approximations. For an L- layer network, let Rm=max |x(()) R,= 1a0) R²=E, |=(0), and W,=E; | Only our approximation scales linearly with both the output resolution and parameters.
Block diag.
O W
O(R W
KFAC
O + W))
ORR + W
Exact diag.
O
O m + W.)
O + W.)
Approx. diag. (ours)
( + W.)
O + W.)
Mixed diag. (ours)
O Rm W.))
O + W.)
+ +
MJ
(a) Block diagonal
(b) Exact diagonal
(c) Approx. diagonal
(d) Mixed diagonal
Figure 5: Comparison of Hessian approximation methods. Common approximations (a-b) scale quadratically with the output resolution. Our proposed approximate and mixed diagonal Hessians (c-d) scale linearly with the resolution. This is essential for scaling the LAE to large images.
input of each layer. This process leads block diagonal approximation of the Hessian as illustrated in Fig. 5(a). However, diagonal blocks are generally too large to store and invert. To combat this, each block can be further approximated by exact diagonal (Daxberger al., 2021) as depicted in Fig. 5(b). This scales linearly w.r.t. parameters, but still scales quadratically w.r.t. the output resolution (Tab. 1).
0,
To scale our Laplacian autoencoder to data, propose to approximate the diagonal of the Hessian rather than relying on exact computations. This is achieved by only backpropagating diagonal form of M as illustrated in Fig. 5(c). This assumes that features from the same layer are uncorrelated and consequently have linear complexity in both time and memory with respect the output dimension (Tab. 1). This makes it viable for our model.
l0 Jn (1
M
11
M0
Figure 6: Forward pass of feature map ir for layer with parameters ( and extended backward pass in which M is backpropagated to previous layers. Via the chain rule and M the Hessian of each layer can be computed efficiently.
We can further tailor this approximation to the autoencoder setting by leveraging the bottleneck architecture. We note that the quadratic scaling of the exact diagonal Hessian is less of an issue in the layers near the bottleneck than in the layers closer to the output space. We can therefore dynamically switch between our approximate diagonal and the exact one, depending on the feature dimension. This lessens the approximation error while remaining tractable practice. We provide more details on the fast hessian computations in Appendix E.
4 Related Work
Deep generative models, and particularly the family of variational auto-encoders (VAES) (Kingma and Welling, 2014; Rezende et al., 2014), address unsupervised representation learning from probabilistic viewpoint by approximating the posterior over the representation space. Despite their widespread adoption, model parameters are still deterministic and sensitive to ill-suited local minima, e.g. over-fitted to training data (Zhang et al., 2021), which may cause poor generalization. The Bayesian NNs favour inference over the NN weights for addressing such issues (MacKay. 1995; Neal, 1996). This approach deduces distributions on data space by learning posterior distributions on the parameter space. However, several shortcomings (Wenzel al., 2020), e.g. expensive training,

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Approx
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Time [sec]
40
KFAC
20
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Input size
Figure 7: Memory time usage of Hessian approximations. The exact and KFAC scales poorly with the image resolution. In contrast, our proposed approximate diagonal Hessian scales linearly.
tuning, and implementations, often limit their applicability to autoencoder-style models. Alternatively, other methods such deep ensembles (Lakshminrryyanan al., 2017), stochastic weight averaging (SWAG) (Maddox et al., 2019) or Monte-Carlo dropout (Gal and Ghahramani, 2016) also promise Bayesian approximations to NN weight's posterior, but at the cost of increased training time, poor empirical performance or limited Bayesian interpretation.
As demonstrated by Daxberger al. (2021) LA is scalable and well-behaved alternative to Bayesian NNs if used post-hoc to approximate the intractable posterior over the weights after maximun-a- posteriori (MAP) training in classification and regression. The general utility of LA has also motivated its use as an approximation to the marginal likelihood over NN weights. Recent methods, including Daxberger al. (2021), have explored this path find model hyperparameters (Immer et al., 202 or learning invariances (Immer et al., 2022). However, its computational burden, for instance in Hessian matrices, has prescribed diagonal or Kronecker factored approximations (Ritter al., 2018; Martens and Grosse, 2015a; Botev et al., 2017), which are now widely used for second-order optimization. We provide more details on the connections to existing hessian based methods Daxberger al. (2021); Zhang et al. (2017): Khan et al. (2017), Bayes by Backpropagation Blundell et al. (2015) and Adam Kingma and Ba (2015a) in Appendix C.
The full Bayesian perspective on VAE weights was first explored by Daxberger and Hernández- (2019) which we find similar in spirit to work. In contrast to them (1) follow principled Bayesian derivation.
(2) Neither do we depend on Hamiltonian Monte- Cateo sampling, which is generally hard to scale efficient training.
5 Experiments
First, we demonstrate the computational advantages of the proposed Hessian approximation, and then that our sampling-base based training leads well-calibrated uncertainties that can be used for OOD detection, data imputation, and semi-supervised learning. For all the downstream tasks we consider the following baselines: AE (Hinton and Salakhutdinov, 2006) with constant and learned variance, VAR (Rezende et al., 2014; Kingma and Welling, 2014), Monte-Carke dropout AE (Gal and Ghahramani, 2016) and Ensembles of AE (Lakshminarayaaan et al., 2017). We extend StochMan et al. (2021) with the Hessian backpropagation for the approximate and mixed diagonals. The training code implemented PyTorch and available². Appendix A provides more details on the experimental setup.
Efficient Hessian Approximation. For practical applications, training time and memory usage the Hessian approximation must be kept low. We here show that the proposed approximate diagonal Hes- sian is sufficient and even outperforms other approximations when combined with our online training.
Fig. 7 show the time and memory requirement for different approximation methods as a function of input size for 5-layer convolutional network that preserves channel and input dimension. As baselines we use efficient implementations of the exact and KFAC approximation (Daxberger et al., 2021; Dangel et al., 2020). The exact diagonal approximation run out of memory for ~ 36 x 36 image on a 11 Gb NVIDIA GeForce GTX 1080 Ti. In contrast, our approximate diagonal Hessian scales linearly with the resolution, which is especially beneficial for convolutional layers.
2 https: / //tpt/gb.tttttt/////egtg.. http:///eeeeeeeeeeeaaar/r//a//ae
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7
7
Hessian
-logp(z)) 4
MSE 4
KFAC
9683.9 + 2455.0
121.6 24.5
Exact
283.3 + 88.6
27.1
Approx
232.0 - 65.5
26.6 0.6
Exact"
25.8 + 0.2
25.7
Approx
25.9 + 0.4
25.8 + 0.4
Table 2: Online training (indicated by ") outper- forms post-hoe L.A. The approximate diagonal has similar performance to the exact diagonal for both post-hoc and online LA.
Approx
Exact
KFAC
Approx
Exact
(a) Poss-boe
(b) Onlize
Figure 8: Mean and variance of 100 sampled NN.
Tab. 2 shows that the exact or approximate Hessian diagonal has similar performance for both post-hoc and online training. Using post-hoc LA results in good mean reconstructions (low MSE), each sampled NN does not give good reconstructions (low log p(x)). Using our online training procedure results in a much higher bog-likelihood. This indicates that every sampled NN predicts good reconstructions.
Fig. 8 shows the latent representation, mean, and variance the reconstructions with the KFAC, exact and approximate diagonal for both post-hoc and online setup. Note that the online training makes the uncertainties better fitted, both in latent and data space. These well-fitted uncertainties have several practical downstream applications, which we demonstrate next.
Out-of-Distribution (OoD) Detection capabilities are critical for identifying distributional shifts, outliers, and irregular inputs, which can hinder the propagation of erroneous decisions in automated system. We evaluate OoD performance on the commonly used benchmarks (Nalisnick et al., 2019b), where use FASHIONMNISS
(Xiao al., 2017) in-distribution and MNIST (Lecun et al., 1998) as OOD. Fig. 9 (c) shows that our online LAE outperforms existing models in both ligeelkl and Typicality score (Nalisnick et al., 2019a). This stems from the calibrated model uncertainties, which are exemplified in the models ability to detect OoD examples from the uncertainty deduced in latent and output space; see Fig. (a.b) for ROC curves.
Fig. 10 shows distribution of the output variances for in- and OoD data. This illustrates that using LA improves OoD detection. Furthermore, the online training improves the model calibration.
Method
logp(ı)
Typicality
o output
latent
AE MSE
0.98
0.98
AE likelihood
0.66
0.70
0.65
VAE
0.54
0.62
0.46
0.5
MC-AE
0.97
0.97
0.08
0.31
LAE (post-hoe)
0.98
0.98
0.74
0.96
LAE (online)
0.98
0.98
0.91
0.96
(a) Latent
(b)
2
output
(c) AUROC (t) for multiple statistics.
Figure 9: Out of Distribution detection. In-distribution data FashionMnist and OoD data Mnist. (a) and (b) shows the ROC curves for latent and output space uncertainties. (c) shows (AUROC for log- likelihood, typicality score, latent space Jjstent and output space Jusput uncertainties. Online LAE able discriminate between in and OoD using the deduced variances in latent and output space.
ID OoD
(a) AE Hislibood
(b) MC-AE
(c) VAB
(c) LAE (pust-huc)
(e) LAE (online)
Figure 10: Histograms of variance for in- and OoD reconstructions in output space. Note that MC- separates the distributions well, but model assigns higher variance to the than OOD data.

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(a) VAE
(b) LAE (post-hoc)
(c) LAB (online)
Figure 11: Missing data imputation & generative capabiliiies. Online training of LAE improves representational robustness. This is exemplified by the multimodal behavior in the data imputation (top rows) that accurately model the ambiguity in the data. The bottom two rows show that the L.A.F able to generate crisp digits from random noise.
Method
MSE +
log f
Acc. f
ECE +
MCE 4
RMSCE +
Classifier
0.53
0.16
0.25
0.18
VAE
104.73
-104.75
0.22
0.18
0.34
0.19
MC-AE
106.05
-106.05
0.45
0.28
0.38
0.29
AE Ensemble
96.23
-100.94
0.53
0.12
0.2
0.13
LAE (post-hoc)
101.62
-107.25
0.51
0.16
0.27
0.18
LAE (online)
99.59
-106.29
0.53
0.12
0.16
0.13
Table 3: Reconstruction quality measured by the MSE and log- likklihheood for the data imputation.
Our well-calibrated uncertainties propagates to the MNIST classifier and improves the calibration metrics ECE, MCE and RMSCE.
Missing Data imputation. Another application of stochastic representation learning is to provide distributions over unobserved values (Rezende et al., 2014). In many application domains, sensor readings go missing, which we may mimic by letting parts of an image unobserved. Rezende et al. (2014) show that the observed part, by imputing the missing pixels with noise and repeatedly encode and decode while keeping observed pixels fixed. Fig. show samples using this procedure from a VAE, post-hoc LAB and our online LAE, where we only observe the lower half of an MNIST image. This implies ambiguity about the original digit, e.g. the lower half of "5" could be "3" and similarly "7" could be a "9". Our L.A captures this ambiguity, which is exemplified by the multi-modal reconstructions from the sampled networks Fig. 11. The baselines only capture unimodal reconstructions.
then draw samples from the distribution of the entire image conditioned
Capturing the ambiguity partly missing data can improve downstream tasks such as the calibration of an image classifier. In Fig. 11 (c) we demonstrate how averaging the predictions of simple classifier across reconstructions improves standard calibration metrics. This because the classifier inherits the uncertainty and ambiguity in the learned representations. A deep ensemble of AES performs similarly to ours, but comes at the cost of training and storing multiple models.
When the entire input image is missing, the imputation procedure can be seen sampling mecha- nism, such that our LAB can viewed as a generative model. The bottom rows in Fig. 11 show that the I.AE indeed does generate sharp images from multi-modal distribution.
Attrihute
AE
VAE
MC-AB
LAE
Arched Eyebrows
0.50
0.52
0.55
0.60
Attractive
0.52
0.50
0.49
0.53
Accurscy
MC-AE
Bald
0.98
0.98
0.98
0.98
VAE (online)
Wearing Lipstick
0.52
0.49
0.50
0.54
Heavy Makeup
0.45
0.52
0.49
0.56
Overall
0.73
0.72
0.73
0.74
Number labels
Table 4: Semi-supervised classification accuracy on CELEBA using only 10 labeled datapoints. refers online LAE.
Figure 12: Accuracy as function of the number labels per class on MNIST.

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Figure 13: Sample reconstructions on CELEBA. The top row shows the mean reconstruction and the bottom row shows the variance of the reconstructed images.
Semi-supervised learning combines small amount of label data with large amount of unlabeled data. The hope is that the structure the unlabeled data can be used to infer properties of the data that cannot be extracted from few labeled points. Embedding the same labeled data point multiple times using a stochastic representation scales up the amount of labeled data that is available during training.
Fig. 12 shows the accuracy of K K-nearest neighbor classifier trained on different amounts of labeled data from the MNIST dataset. For all models with a stochastic encoder, we encode each labeled datapoint 100 times and repeat the experiment 5 times. When only few labels per class are available (1-20) we clearly observe that our LAE model outperforms all other models, stochastic and deterministic. Increasing the number of labels beyond 100 per class makes the AE and LAE equal in their classification performance with the AE model eventually outperforming the LAE model.
In Tab. 4 we conduct similar experiment on the CELEBA (Liu et al., 2015) facial dataset, where the the task is to predict 40 different binary labels per data point. When evaluating the overall accuracy of predicting all 40 facial attributes, we see no significant difference in performance. However, when we zoom in on specific facial attributes we gain a clear performance advantage over other models. Fig, 13 shows the mean and variance of five reconstructed images. The online LAE produces well-calibrated uncertainties in the output space and scales to large images.
Limitations. Empirically, the LAE improvements are more significant for overparameterized net- works. The additional capacity propriate. It seems the regularization induced by marginalizing 0 compensates for the added flexibility.
help the optimizer find local mode where Gaussian fit is ap-
6 Conclusion
In this paper. we have introduced Bayesian autoencoder that is realized using Laplace approxi- mations. Unlike current models, this Laplacian autoencoder produces well-behaved uncertainties both latent and data space. We have proposed a novel variational lower-bound of the autoencoder evidence and an efficient way to compute its Hessian on high dimensional data that scales linearly with data size. Empirically, we demonstrate that our proposed model predicts reliable stochastic detection,
representations that are useful for multitude of downstream tasks: out-of-distribute missing data imputation, and semi-supervised classification. Our work opens the way for fully Bayesian representation learning where we can marginalize the representation in downstream tasks. We find this to consistently improve performance.
Acknowledgments and Disclosure of Funding
This work was supported by research grants (15334, 42062) from VILLUM FONDEN. This project has also received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement 7573360)) This work was funded in part by the Novo Nordisk Foundation through the Center for Basic Machine Learning Research in Life Science (NNF200C0062606) and by the Pioneer Centre for AI DNRF grant number Pl.
10

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Checklist
1. For all authors...
(a) Do the main claims made the abstract and introduction accurately reflect the paper's contribu- tions and scope? [Yes]
(b) Did you describe the limitations your work? [Yes]
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(a) Did you state the full set of assumptions of all theoretical results? [Yes] In Section 2 and the
Appendix.
(b) Did you include complete proofs of all theoretical results? [Yes] See appendix.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as URL)? [Yes] See appendix.
(h) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See appendix.
(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No]
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14


Reviews:
[{"review_id": "W97JZ6uVO5E", "reviewer": "Reviewer_meVs", "summary": "This paper introduces a novel Bayesian autoencoder. By applying Monte Carlo EM with a variational distribution with Laplacian approximation, the proposed model become tractable and can perform iterative updates. To make the proposed method scalable to large datasets, the authors also propose to approximate the Hessian matrix to make the running time scales linearly with the input size. Experiment results show that the proposed method perform well on tasks like OOD detection and data imputation.", "questions": "1. In Table 2, why the exact method (with post-hoc) performs worse than the approximation method (in both the log-likelihood and MSE)? Are the metric diffs big enough to say that the approximation method performs better here than the exact method? If so, it seems weird and it will be good if the authors can check the implementations. If not, it will be better if the authors can show the confidence intervals of these metrics.\n\n2. The authors proposed to apply diagonal approximation for the Hessian. This is definitely good for large scale training. However, a diagonal approximation is probably an over approximation. Did you try some idea with (potentially) tighter approximations?", "limitations": "This paper discussed about the limitations of their work.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n\n1. The idea of applying LA and Monte Carlo EM is reasonable. Through these applications, the neural networks can be sampled and the updates for the parameters become a normal set of updates for posterior computations, which is tractable.\n\n2. The proposed updates can ensure the Hessian part is always negative definite, which makes the process smooth.\n\n3. The Hessian diagonal approximation makes sure that the proposed method can scale up to higher dimensionality.\n\nWeaknesses:\n\n1. This paper is hard to follow and many details are missing or hard to find. For example, the second paragraph of Section 1.1 is unclear. It also has some confusing presentations (e.g. the first sentence says that $\\sigma^2(z)$ should be as large as possible to make $p(x)$ large, it seems like this is not the case) or some typos (e.g. Eq 1 is a lower bound on $\\log p(x)$, not $p(x)$ as shown there). Another example is Figure 1, where no detailed introduction about the definition of this figure was shown in the caption. It will be better if the presentation can be improved.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "CH7aWkS6KL", "reviewer": "Reviewer_4XuV", "summary": "The authors propose to use Bayesian autoencoders for unsupervised representation learning, where they approximate the posterior over the parameters using a Laplace approximation (LA). In contrast to previous work, they update the LA during training, for which they devise a Monte Carlo EM iterative update scheme. They show that this approach performs really well on a range of different problems.", "questions": "Overall, this is a really nice paper and I don't have much to criticize. Just a few minor comments below.\n\n- l. 24: \"VAEs\" -> \"VAE's\"\n- Fig. 1 caption: \"laplace\" -> \"Laplace\"\n- l. 136: The fact that the GGN is the exact Hessian in the linearized model has also been noted by Immer et al. [1]\n- l. 170: \"sensible\" -> \"sensitive\"\n\n\n[1] https://proceedings.mlr.press/v130/immer21a.html", "limitations": "The authors have addressed all limitations and ethical concerns.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n\n- The paper is well written and the contributions are clear.\n- The Monte Carlo EM update seems like an elegant solution for training with Laplace approximations.\n- The empirical performance of the proposed method is impressive.\n\nWeaknesses:\n\n- A few typos (see below)", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "77kAF_Soiry", "reviewer": "Reviewer_vo1U", "summary": "The authors propose a novel framework for learning stochastic representations using a Bayesian autoencoder called Laplacian Autoencoders. They derive a novel training objective to train this Bayesian autoencoder and propose a novel approximation for the fast computation of the Hessian that makes their approach viable on high-dimensional datasets. The experiments section demonstrates the advantage of this approach over previous ones (notably over a post-hoc analysis of a pretrained autoencoder) on tasks like out-of-distribution detection, missing data imputation or semi-supervised learning.", "questions": "Here are some suggestions that could help:\nI would suggest to find a way to better detail the derivation of p. 3: conditionning on f is hard to grasp and I had troubles following all the details. Some ideas might need to be justified as they can appear surprising for a non expert of the domain (e.g. using the Hessian of the lower bound instead of the true term; the introduction of the $\\alpha$ parameter).\n\nThe caption of Fig. 1 could precise what is shown.\nThere are some typos: many concerning missing s on the third person (e.g. ll. 58, 128, 205, 212),  what is the difference between $f^{(t)}$, $f^{(t)}_{\\theta_t}$, $f_{\\theta_t}$?", "limitations": "The authors adequately addressed the limitations and potential negative societal impact of their work.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This paper is very well-written: both the writing and the presentation are precise and clear. It is well illustrated and the experiments show the relevance of this approach on a wide variety of tasks. The problem is properly introduced via a well-chosen example and the related works section properly discusses the  relevant literature.\nI still had some troubles with pp. 3 and 4 due to some notations issues or quick derivations.\nThe problem tackled will interest many researchers in the community and this article proposes both a clear and principled derivation of the objective together with practical considerations.\n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept