Data is often gathered sequentially in the form of a time series, which consists of sequences of data points observed at successive time points. Dynamic time warping (DTW) defines a meaningful similarity measure between two time series. Often times, the pairs of time series we are interested in are defined on different spaces: for instance, one might want to align a video with a corresponding audio wave, potentially sampled at different frequencies. In this work, we propose *Gromov Dynamic Time Warping* (GDTW), a distance between time series on potentially incomparable spaces, and apply it to various settings, including barycentric averaging, generative modeling, and imitation learning.

# Time Series Alignment

Sakoe and Chiba^{1} consider the problem of aligning two time series `$\boldsymbol{x} \in \mathcal{X}^{T_x}$`

and `$\boldsymbol{y} \in \mathcal{Y}^{T_y}$`

, where potentially `$T_x \neq T_y$`

. This is formalized as

```
$$
\operatorname{DTW}(\boldsymbol{x},\boldsymbol{y}) = \min_{\mathbf{A} \in \mathcal{A}(T_x,T_y)} \langle\mathbf{D}, \mathbf{A}\rangle_{\operatorname{F}}
$$
```

where `$D_{ij} = d_\mathcal{X}(x_i,y_j)$`

is the pairwise distance matrix and `$A_{ij}$`

is `$1$`

if `$x_i$`

and `$y_j$`

are aligned, and `$0$`

otherwise. A DTW alignment of two time series is shown in Figure 1.

A practical drawback of DTW is the need for both time series `$\boldsymbol{x}$`

and `$\boldsymbol{y}$`

to live on the same spaces, with a metric `$d_{\mathcal{X}}$`

. This can cause the following issues.

- Alignment can fail for time series that are only close up to isometries, such as rotations and translations.
- The method doesn’t apply to time series which are defined on different spaces, such as location coordinates for
`$\boldsymbol{x}$`

and pixel values for`$\boldsymbol{y}$`

.

In such cases, defining a meaningful distance between samples from the two sequences is impractical as it would require detailed understanding of the objects we wish to study.

# Dealing with Incomparable Spaces

Motivated by connections between DTW and optimal transport,^{2} we introduce a distance between time series `$\boldsymbol{x} \in \mathcal{X}^{T_x}$`

and `$\boldsymbol{y} \in \mathcal{Y}^{T_y}$`

defined on potentially incomparable metric spaces. The key idea is to define a loss function which compares pairwise distances in `$\mathcal{X}^{T_x}$`

with those in `$\mathcal{Y}^{T_y}$`

. For this, we define the Gromov dynamic time warping distance as

```
$$
\operatorname{GDTW}(\boldsymbol{x},\boldsymbol{y})=\min_{\mathbf{A} \in \mathcal{A}(T_x,T_y)} \sum_{ijkl} \mathcal{L} \big(d_{\mathcal{X}}(x_i,x_k),d_{\mathcal{Y}}(y_j,y_l)\big) A_{ij}A_{kl},
$$
```

where `$d_{\mathcal{X}}$`

is a distance defined on `$\mathcal{X}$`

, and `$d_{\mathcal{Y}}$`

a distance defined on `$\mathcal{Y}$`

. We solve the optimization problem over the set of alignment matrices by applying a Frank–Wolfe-inspired algorithm. Results can be seen in Figure 2 for different rotations and translations of the original time series.

Similarly to DTW, GDTW suffers from unpredictability when the time series is close to a change point of the optimal alignment matrix because of the discontinuity of derivatives. To remedy this, we introduce a softened version of this expression, mirroring the definition of soft DTW.^{3} This allows smoother derivatives when applying it to for instance generative modeling of time series and imitation learning.

# Applications

We showcase a number of applications of Gromov DTW. We considered 3 settings: barycentric averaging, generative modeling, and imitation learning.

**Barycentric averaging**: we extend the algorithm of Peyré et al.^{2}to the sequential setting via coordinate descent on the GDTW objective. We plot barycenters under several warping approaches in Figure 3.**Generative modeling**: we extend the algorithm of Genevay et al.^{4}and Bunne et al.^{5}to the sequential setting by leveraging GDTW as ground metric in an entropic Wasserstein objective. Samples can be observed in Figure 4.**Imitation learning**: We propose an approach to performing imitation learning when the agent and expert do not live on comparable spaces, which consists in minimizing the Gromov-DTW between expert demonstrations and agent rollouts. This is illustrated in Figure 5.

# Summary

We introduce a distance between time series living on potentially incomparable spaces, Gromov DTW, which significantly broadens the range of applications of previous time-series metrics like DTW. We also propose a smoothed version that alleviates the discontinuity of GDTW’s gradient. Gromov DTW is a general concept for comparing two time series and can be applied to a number of applications, including barycentric averaging, generative modeling and imitation learning.

# References

^{1}

H. Sakoe, S. Chiba. Dynamic Programming Algorithm Optimization for Spoken Word Recognition. ICASSP, 2018.

^{2}

G. Peyré, M. Cuturi, J. Solomon. Gromov–Wasserstein Averaging of Kernel and Distance Matrices. ICML, 2016.

^{3}

M. Cuturi, M. Blondel. Soft-DTW: A Differentiable Loss Function for Time-Series. ICML, 2017.

^{4}

A. Genevay, G. Peyré, M. Cuturi. Learning Generative Models with Sinkhorn Divergences. AISTATS, 2018.

^{5}

C. Bunne, D. Alvarez-Melis, A. Krause, S. Jegelka. Learning Generative Models Across Incomparable Spaces. ICML, 2019.