Measuring the distance between data points is a central ingredient of many data analysis and machine learning applications. Several kernel methods (KernelPCA (Schölkopf et al., 1997), KernelNMF (Li & Ding, 2006), etc.), and other non-parametric approaches such as k-nearest neighbours (Altman, 1992)
rely on the availability of a suitable distance function. Computer vision pipelines, e.g. tracking over time, perform matching based on similarity scores.
But designing a distance function can be challenging: it is not always obvious to write down mathematical formulae that accurately express a notion of similarity. Learning such functions has hence been proven as a viable alternative to manual engineering in this respect (NCA (Goldberger et al., 2005), metric learning (Xing et al., 2003), etc.). Often, these methods rely on the availability of pairs labelled as similar or dissimilar. A different route is that of exploiting the structure that latent-variable models learn. The assumption that a set of high-dimensional observations is explained by points in a much simpler latent space underpins these approaches. In their respective probabilistic versions, a latent prior distribution is transformed non-linearly to give rise to a distribution of observations. The hope is that simple distances, such as the Euclidean distance measured in latent space, implement a function of similarity. Yet, these approaches do not incorporate the variation of the observations with respect to the latent points. For example, the observations will vary much more when a path in latent space will cross a class boundary.
In fact, recent approaches to non-linear latent variable models, such as the generative adversarial network (Goodfellow et al., 2014) or the variational auto-encoder (VAE) (Kingma & Welling, 2014; Rezende et al., 2014), regularise the latent space to be compact, i.e. to remove low-density regions. This is in contrast to the aforementioned hope that Euclidean distances appropriately reflect similarity.
The above insight leads us to the development of flat manifold variational auto-encoders. This class of VAEs defines the latent space as Riemannian manifold and regularises the Riemannian metric tensor to be a scaled identity matrix. In this context, a flat manifold is a Riemannian manifold, which is isometric to the Euclidean space. To not compromise the expressiveness, we relax the compactness assumption and make use of a recently introduced hierarchical prior (Klushyn et al., 2019). As a consequence, the model is capable of learning a latent representation, where the Euclidean metric is a proxy for the similarity between data points. This results in a computational efficient distance metric which is practical for applications in real-time scenarios.
2 Variational Auto-Encoders with Flat Latent Manifolds
2.1 Background on Learning Hierarchical Priors in VAEs
Latent-variable models are defined as
where represents latent variables and the observable data. The integral in Eq. (1) is usually intractable but it can be approximated by maximising the evidence lower bound (ELBO) (Kingma & Welling, 2014; Rezende et al., 2014):
where is the empirical distribution of the data . The distribution parameters of the approximate posterior and the likelihood
are represented by neural networks. The prior
is usually defined as a standard normal distribution. This model is commonly referred to as the variational auto-encoder (VAE).
However, a standard normal prior often leads to an over-regularisation of the approximate posterior, which results in a less informative learned latent representation of the data (Tomczak & Welling, 2018; Klushyn et al., 2019). To enable the model to learn an informative latent representation, Klushyn et al. (2019) propose to use a flexible hierarchical prior , where is the standard normal distribution. Since the optimal prior is the aggregated posterior (Tomczak & Welling, 2018), the above integral is approximated by an importance-weighted (IW) bound (Burda et al., 2015) based on samples from . This leads to a model with two stochastic layers and the following upper bound on the KL term:
where is the number of importance samples. Since it has been shown that high ELBO values do not necessarily correlate with informative latent representations (Alemi et al., 2018; Higgins et al., 2017)—which is also the case for hierarchical models (Sønderby et al., 2016)—different optimisation approaches have been introduced (Bowman et al., 2016; Sønderby et al., 2016). Klushyn et al. (2019) follow the line of argument in (Rezende & Viola, 2018) and reformulate the resulting ELBO as the Lagrangian of a constrained optimisation problem:
with the optimisation objective , the inequality constraint , and the Lagrange multiplier . is defined as the reconstruction-error-related term in . Thus, we obtain the following optimisation problem:
Building on that, the authors propose an optimisation algorithm—including a -update scheme—to achieve a tight lower bound on the log-likelihood. This approach is referred to as variational hierarchical prior (VHP) VAE.
2.2 Learning Flat Latent Manifolds with VAEs
The VHP-VAE is able to learn a latent representation that corresponds to the topology of the data manifold (Klushyn et al., 2019). However, it is not guaranteed that the (Euclidean) distance between encoded data in the latent space is a sufficient distance metric in relation to the observation space. In this work, we aim to measure the distance/difference of observed data directly in the latent space by means of the Euclidean distance of the encodings.
Chen et al. (2018a); Arvanitidis et al. (2018) define the latent space of a VAE as a Riemannian manifold. This approach allows for computing the observation-space length of a trajectory in the latent space:
where is the Riemannian metric tensor, and the time derivative. We define the observation-space distance as the shortest possible path
between two data points. The trajectory that minimises is referred to as the geodesic. In the context of VAEs, is transformed by a continuous function —the decoder—to the observation space. The metric tensor is defined as , where is the Jacobian of the decoder.
To measure the observation-space distance directly in the latent space, distances in the observation space should be proportional to distances in the latent space:
where we define the Euclidean distance as the distance metric. This requires that the Riemannian metric tensor is . As a consequence, the Euclidean distance in the latent space corresponds to the geodesic distance. We refer to a manifold with this property as flat manifold (Lee, 2006). To obtain a flat latent manifold, the model typically needs to learn complex latent representations of the data (see experiments in Sec. 4). Therefore, we propose the following approach: (i) to enable our model to learn complex latent representations, we apply a flexible prior (VHP), which is learned by the model (empirical Bayes); and (ii) we regularise the curvature of the decoder such that .
For this purpose, the VHP-VAE, introduced in Sec. 2.1, is extended by a Jacobian-regularisation term. We define the regularisation term as part of the optimisation objective, which is in line with the constrained optimisation setting. The resulting objective function is
where is a hyper-parameter determining the influence of the regularisation and the scaling factor. Additionally, we use a stochastic approximation (first order Taylor expansion) of the Jacobian to improve the computational efficiency (Rifai et al., 2011b):
where is the Jacobian of the -th latent dimension and
a standard basis vector. This approximation method allows for a faster computation of the gradient and avoids the second-derivative problem of piece-wise linear layers(Chen et al., 2018a).
However, the influence of the regularisation term in Eq. (2.2) on the decoder function is limited to regions where data is available. To overcome this issue, we propose to use mixup, a data-augmentation method (Zhang et al., 2018)
, which was introduced in the context of supervised learning. We extend this method to the VAE framework (unsupervised learning) by applying it to encoded data in the latent space. This approach allows augmenting data by interpolating between two encoded data pointsand :
with , and . In contrast to (Zhang et al., 2018), where limits the data augmentation to only convex combinations, we define to take into account the outer edge of the data manifold. By combining mixup in Eq. (11) with Eq. (2.2), we obtain the objective function of our flat manifold VAE (FMVAE):
Inspired by batch normalisation, we define the squared scaling factor to be the mean over the batch samples and diagonal elements of (see App. A.2 for empirical support):
By using augmented data, we regularise to be a scaled identity matrix for the entire latent space enclosed by the data manifold. Hence, the VHP-FMVAE learns an Euclidean latent space. As a consequence, the function (decoder) is—up to the scaling factor —distance-preserving since , where and refer to the distance in the observation and latent space, respectively.
The decoder of the proposed approach satisfies the Lipschitz continuity condition . We consider the decoder function, and hence the latent space as smooth if , where is the Lipschitz constant.
3 Related Work
Interpretation of the VAE’s latent space. In general, the latent space of VAEs is considered to be Euclidean (e.g. Kingma et al., 2016; Higgins et al., 2017), but it is not constrained to be Euclidean. This can be problematic if we require a precise metric that is based on the latent space. Some recent works (Mathieu et al., 2019; Grattarola et al., 2018) adapted the latent space to be non-Euclidean to match the data structure. We solve the problem from another perspective: we enforce the latent space to be Euclidean.
Jacobian and Hessian regularisation. In (Rifai et al., 2011a), the authors proposed to regularise the Jacobian and Hessian of the encoder. However, it is more difficult to augment data in the observation space than in the latent space. Encoder regularisation enables the model to perform better in case of, e.g., object recognition by means of the latent space. By contrast, decoder regularisation enables the model to do tasks such as generating motions based on the latent space. In (Hadjeres et al., 2017), the Jacobian of the decoder was regularised to be as small as possible/zero. On the contrary, we regularise the the Riemannian metric tensor to be a scaled identity matrix, and hence the Jacobian to be constant, and hence the Hessian to be zero. (Nie & Patel, 2019) regularised the Jacobian with respect to the weights for GANs. In terms of supervised learning, (Jakubovitz & Giryes, 2018) used Jacobian regularisation to improve the robustness for classification.
Metric learning. Various metric learning approaches for both deep supervised and unsupervised models were proposed. For instance, deep metric learning (Hoffer & Ailon, 2015) used a triplet network for supervised learning. (Karaletsos et al., 2016) introduced an unsupervised metric learning method, where a VAE is combined with triplets. However, a human oracle is still required. By contrast, our approach is completely based on unsupervised learning, using the Euclidean distance in the latent space as a distance metric. Our proposed method is similar to the metric learning methods such as Large Margin Nearest Neighbour (Weinberger & Saul, 2009), which pulls target neighbours together and pushes impostors away. The difference is that our approach is an unsupervised method.
Constraints in latent space. Constraints on time (e.g. Wang et al., 2007; Chen et al., 2016, 2015) allow obtaining similar distance metrics in the latent space. Additionally, due to the lack of data, constraints on time cannot guarantee that the metric is correct between of different sequences. However, our method can be used for general data-sets.
Data augmentation. The latent space is formed arbitrarily in regions where data is missing. Zhang et al. (2018) proposed mixup, an approach for augmenting input data and labels for supervised learning. Various follow-up studies of mixup were developed, such as (Verma et al., 2018; Beckham et al., 2019). We extend mixup to the VAE framework (unsupervised learning) by applying it to encoded data in the latent space. This facilitates the regularisation of regions where no data is available. As a consequence, similarity of data points can be measured in the latent space by applying the Euclidean metric.
Geodesic. Recent studies on geodesics for generative models (e.g. Tosi et al., 2014; Chen et al., 2018a; Arvanitidis et al., 2018) are focusing on methods for computing/finding the geodesic in the latent space. By contrast, we use the geodesic/Riemannian distance for influencing the learned latent manifold. (Frenzel et al., 2019)
projected the latent space to a new latent space, where the geodesic is equivalent to the Euclidean interpolation. However, these two separate processes—VAEs and projection—probably hinder the model to find the latent features autonomously. Another difference is the assumption of previous work is that distances, defined by geodesics, can only be measured by following the data manifold. This is useful in scenarios such as avoiding unseen barriers between two data points, e.g.,(Chen et al., 2018b), but it does not allow measuring distances between different categories. In this work, we focus on learning a general distance metric.
We test our method on artificial pendulum images, human motion data, MNIST, and MOT16. We measure the performance in terms of equidistances, interpolation smoothness, and distance computation. Additionally, our method is applied to a real-world environment—a video-tracking benchmark. Here, the tracking and re-identification capabilities are evaluated.
The Riemannian metric tensor has many intrinsic properties of a manifold and measures local angles, length, surface area, and volumes (Bronstein et al., 2017). Therefore, the models are quantified based on the Riemannian metric tensor by computing condition numbers and magnification factors. The condition number, which shows the ratio of the most elongated to the least elongated direction, is defined as , where
is the largest eigenvalue of. The magnification factor (Bishop et al., 1997) depicts the sensitivity of the likelihood functions. When projecting from the Riemannian (latent) to the Euclidean (observation) space, the can be considered a scaling coefficient. Since we cannot directly compare the s of different models, the s are normalised/divided by their means. The closer the conditional number and the normalised MF are to one, the more invariant is the model with respect to the Riemannian metric tensor. In other words: the conditional number and the normalised MF are metrics to evaluate whether is approximately constant and proportional to .
In order to make the visualisations of the magnification factor in Sec. 4.1 (Fig. 1) and Sec. 4.2 (Fig. 3 & Fig. 7) comparable, we define the respective upper range of the colour-bar as . and are computed with training data and by using a grid area, respectively.
4.1 Artificial Pendulum Data-set
The pendulum data-set (Klushyn et al., 2019; Chen et al., 2018a) consists of -pixel images generated by a pendulum simulator. We generated images with joint angles in the ranges of degrees. Additionally, we added Gaussian noise to each pixel.
As seen in Fig. 1, without regularisation, the contour lines are denser in the centre of the latent space. The reason is that, in contrast to the VHP-VAE, the regularisation term in the VHP-FMVAE smoothens the latent space ()—visualised by the and the equidistance plots. In Fig. 2, VHP-FMVAE and VAE-VHP are compared in terms of condition number and normalised . In both cases the VHP-FMVAE outperforms the VHP-VAE.
4.2 Human Motion Capture Database
To evaluate our approach on the CMU human motion data-set (http://mocap.cs.cmu.edu), we select five different movements: walking (subject 35), jogging (subject 35), balancing (subject 49), punching (subject 143), and kicking (subject 74). After data pre-processing, the input data is a 50-dimensional vector of the joint angles. Note that the data-set is not balanced: walking, for example, has more data points than jogging.
|Human||VHP-FMVAE||1.02 0.06||0.93 0.03|
|VHP-VAE||1.23 0.20||0.82 0.10|
|MNIST||VHP-FMVAE||1.01 0.08||0.92 0.05|
|VHP-VAE||1.13 0.22||0.70 0.31|
Equidistance plots. In Fig. 3, we randomly select a data point from each class as centres of the equidistance plots. In case of our proposed method, the equidistance plots are homogeneous, while in case of the VHP-VAE, the equidistance contour lines are distorted in regions of high values. Thus, the mapping from latent to observation space learned by the VHP-FMVAE is approximately distance preserving. Additionally, we use the condition number and the normalised to evaluate based on 3,000 random samples. In contrast to the VHP-VAE, both the condition number and the normalised MF values of the VHP-FMVAE are close to one, which indicates that .
Smoothness. We randomly sample 100 pair points and linearly interpolate between each pair. The second derivative of each trajectory is defined as the smoothness factor. Fig. 5 illustrates that the VHP-FMVAE significantly outperforms the VAE-VHP in terms of smoothness. Fig. 6 shows five examples of the interpolated trajectories.
Verification of the distance metric. To verify that the Euclidean distance in the latent space corresponds to the geodesic distance, we approximates the geodesic by using a graph-based approach (Chen et al., 2019)
. The graph of the baseline has 14,400 nodes, which are sampled in the latent space using a uniform distribution. Each node has 12 neighbours. In Fig.3, five geodesics each are compared to the corresponding Euclidean interpolations. Tab. 1 shows the ratios of Euclidean distances in latent space to geodesics distances, as well as the related ratios in the observation space. To compute the ratios, we randomly sampled 100 pairs of points and interpolated between each pair. If the ratio of the distances is close to one, the Euclidean interpolation approximates the geodesic. The VHP-FMVAE outperforms the VAE-VHP.
Influence of the data augmentation and the identity term . Fig. 4 and Fig. 6(a) show the influence of the data augmentation (see Sec. 2.2). Without data augmentation, the influence of the regularisation term is limited to regions where data is available, as verified by the high values between the different movements. As an additional experiment, Fig. 4 and Fig. 6(b) illustrates the influence of the identity term . If we remove it, the regularisation term becomes . As a consequence, the model is not able to learn a flat latent manifold.
The binarised MNIST data-set (Larochelle & Murray, 2011) consists of 50,000 training and 10,000 test images of handwritten digits (zero to nine) with pixels in size.
Both of our evaluation metrics the condition number and the normalisedshow that the VHP-FMVAE outperforms the VAE-VHP (see Fig. 8 and Fig. 9). In contrast to the VHP-VAE, the VHP-FMVAE learns a latent space, where Euclidean distances are close to geodesic distances (see Tab. 1). This indicates that is approximately constant.
4.4 MOT16 Object-Tracking Database
We evaluate our approach on the MOT16 object-tracking database (Milan et al., 2016), which is a large-scale person re-identification data-set, containing both static and dynamic scenes from diverse cameras.
. SORT is a simple online and real-time tracking method, which uses bounding box intersection-over-union (IOU) for associating detections between frames and Kalman filters for the track predictions. It relies on good two-dimensional bounding box detections from a separate detector, and suffers from ID switching when tracks overlap in the image. DeepSORT extends the original SORT algorithm to integrate appearance information based on a deep appearance descriptor, which helps with re-identification in the case of such overlaps or missed detections. The deep appearance descriptor is trained using asupervised cosine metric learning approach (Wojke & Bewley, 2018). The candidate object locations of the pre-generated detections for both SORT, DeepSORT and our method are taken from (Yu et al., 2016). Further details regarding the implementation can be found in App. A.3.
We use the following metrics for evaluation. indicates that the higher the score is, the better the performance is. On the contrary, indicates that the lower the score is, the better the performance is.
IDF(): ID F Score
IDP(): ID Precision
IDR(): ID Recall
FAR(): False Alarm Ratio
MT(): Mostly Tracked Trajectory
PT(): Partially Tracked Trajectory
ML(): Mostly Lost Trajectory
FP(): False Positives
FN(): False Negatives
IDs(): Number of times an ID switches to a different previously tracked object
MOTA(): Multi-object tracking accuracy
MOTP(): Multi-object tracking precision
MOTAL(): Log tracking accuracy
Tab. 2 shows that the performance of the proposed method is better than that of the model without Jacobian regularisation, and even close to the the performance of supervised learning. All methods depend on the same underlying detector for object candidates, and identical Kalman filter parameters. Compared to baseline SORT which does not utilise any appearance information, DeepSORT has 2.54 times, VHP-VAE-SORT has 2.14 times, VHP-FMVAE-SORT () has 2.41 times and VHP-FMVAE-SORT () has 2.48 times fewer ID switches. Whilst the supervised DeepSORT descriptor has the least, using unsupervised VAEs with flat decoders has only 2.2% more switches, without the need for labels. Furthermore, by ensuring a quasi-Euclidean latent space, one can query nearest-neighbours efficiently via data-structures such as kDTrees. Fig. 10 shows an example of the results. In other examples of the videos, the VHP-FMVAE-SORT works similar as the DeepSORT. Videos of the results can be downloaded at: http://tiny.cc/0s71cz
In this paper, we have proposed a novel approach, which we call flat manifold variational auto-encoder. We have shown that this class of VAEs learns a latent representation, where the Euclidean metric is a proxy for the similarity between data points. This is realised by interpreting the latent space as a Riemannian manifold and by combining a powerful empirical Bayes prior with a regularisation method that constrains the Riemannian metric tensor to be a scaled identity matrix. Experiments on several datasets have shown the effectiveness of our proposed algorithm for measuring similarity. In case of the MOT16 object-tracking database, the performance of our unsupervised method nears that of state-of-the-art supervised approaches.
Thanks to Botond Cseke and Alexandros Paraschos for the useful feedback of this work.
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Appendix A Appendix
a.1 Vector Field
a.2 Influence of
a.3 Implementation of VHP-FMVAE-SORT
We evaluate the performance of our model by replacing the appearance descriptor from DeepSORT with the latent space embedding from the various auto-encoders used, using the same size of 128. The hyperparameters used were held constant: the minimum detection confidence of, NMS max overlap of , max cosine distance , max appearance budget . We tested a VHP-FMVAE, and our regularised VHP-FMVAE with and .
a.4 Optimisation Process
As introduced in (Klushyn et al., 2019), we apply the following -update scheme:
where is defined as
is the Heaviside function and a slope parameter.
a.5 Model Architectures
FC 256, 256. ReLU activation.
|FC 256, 256. ReLU activation. Gaussian.|
|FC 256, 256, ReLU activation.|
|FC 256, 256, ReLU activation.|
|Others||= 0.025, = 1, = 16, .|
|FC 256, 256, 256, 256. ReLU activation.|
|FC 256, 256, 256, 256. ReLU activation. Gaussian.|
|FC 256, 256, 256, 256, ReLU activation.|
|FC 256, 256, 256, 256, ReLU activation.|
|Others||= 0.03, = 1, = 32, .|
|FC 256, 256, 256, 256. ReLU activation.|
|FC 256, 256, 256, 256. ReLU activation. Bernoulli.|
|FC 256, 256, 256, 256. ReLU activation.|
|FC 256, 256, 256, 256. ReLU activation.|
|others||= 0.245 , = 1, = 16, .|
|VGG16 (Simonyan & Zisserman, 2015)|
|Conv2DT+Conv2D 256, 128, 64, 32, 16.|
|ReLU activation. Gaussian.|
|FC 512, 512. ReLU activation.|
|FC 512, 512. ReLU activation.|
|others||= 0.8 , = 1, = 8, .|