Soft-DTW: a Differentiable Loss Function for Time-Series. Appendix material
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1 : a Differentiable Loss Function for Time-Series Appendix material A. Recursive forward computation of the average path matrix The average alignment under Gibbs distributionp γ can be computed with the following forward recurrence, which mimics closely Bellman s original recursion. For eachi n,j m, define E i+,j+ = [ ] e δ i+,j+/γ E i,j i T j e ri+,j+/γ + [ ] [ e δ i+,j+/γ E i,j+ T j e ri+,j+/γ + e δi+,j+/γ E i+,j ] i e rij/γ Here terms r ij are computed following the recursion in Algorithm. Border matrices are initialized to, except for E, which is initialized to[]. Upon completion, the average alignment matrix is stored ine n,m. The operation above consists in summing three matrices of size (i +,j + ). There are exactly (nm) such updates. A careful implementation of this algorithm, that would only store two arrays of matrices, as Algorithm only store two arrays of values, can be carried out innmmin(n,m) space but it would still require(nm) operations.
2 : a Differentiable Loss Function for Time-Series B. Barycenters obtained with random initialization ( =) ( =.) (a) CBF ( =) ( =.) (b) Herring ( =) ( =.) (c) Medical Images ( =) ( =.) (d) Synthetic Control ( =) ( =.) (e) Wave Gesture Library Y
3 : a Differentiable Loss Function for Time-Series C. Barycenters obtained with mean initialization ( =) ( =.) (a) CBF ( =) ( =.) (b) Herring ( =) ( =.) (c) Medical Images ( =) ( =.) (d) Synthetic Control ( =) ( =.) (e) Wave Gesture Library Y
4 D. More interpolation results : a Differentiable Loss Function for Time-Series Left: results obtained under loss. Right: results obtained under soft-dtw (γ = ) loss (a) ArrowHead (b) ECG 5 5 (c) ItalyPowerDemand (d) TwoLeadECG
5 : a Differentiable Loss Function for Time-Series E. Clusters obtained byk-means under DTW or soft-dtw geometry CBF dataset Cluster (8 points) Cluster (9 points) Cluster ( points) (a) (γ =, random initialization) Cluster (8 points) Cluster (8 points) (b) (γ =, mean initialization) Cluster ( points) Cluster (8 points) Cluster ( points) Cluster ( points) (c) (random initialization) Cluster ( points) Cluster ( points) (d) ( mean initialization) Cluster ( points)
6 : a Differentiable Loss Function for Time-Series ECG dataset Cluster (59 points) Cluster ( points) (a) (γ =, random initialization) Cluster (8 points) Cluster (9 points) (b) (γ =, mean initialization) Cluster (8 points) Cluster (7 points) 6 8 (c) (random initialization) 6 8 Cluster (76 points) Cluster ( points) 6 8 (d) ( mean initialization) 6 8
7 F. More visualizations of time-series prediction : a Differentiable Loss Function for Time-Series (a) CBF (b) ECG (c) ECG (d) ShapesAll (e) uwavegesturelibrary Y
8 : a Differentiable Loss Function for Time-Series G. Barycenters: DTW loss (Eq. withγ = ) achieved with random init Dataset γ = γ =. γ =. γ =. Subgradient method mean 5words Adiac ArrowHead Beef BeetleFly BirdChicken CBF Car ChlorineConcentration CinC ECG torso Coffee Computers Cricket X Cricket Y Cricket Z DiatomSizeReduction DistalPhalanxOutlineAgeGroup DistalPhalanxOutlineCorrect DistalPhalanxTW ECG ECG ECGFiveDays Earthquakes ElectricDevices FISH FaceAll FaceFour FacesUCR FordA FordB Gun Point Ham HandOutlines Haptics Herring InlineSkate N/A N/A N/A N/A ItalyPowerDemand MedicalImages MiddlePhalanxOutlineAgeGroup MiddlePhalanxOutlineCorrect MiddlePhalanxTW MoteStrain NonInvasiveFatalECG Thorax NonInvasiveFatalECG Thorax OSULeaf OliveOil PhalangesOutlinesCorrect Phoneme Plane ProximalPhalanxOutlineAgeGroup ProximalPhalanxOutlineCorrect ProximalPhalanxTW RefrigerationDevices ScreenType ShapeletSim ShapesAll SmallKitchenAppliances SonyAIBORobotSurface SonyAIBORobotSurfaceII StarLightCurves Strawberry SwedishLeaf Symbols ToeSegmentation ToeSegmentation Trace TwoLeadECG Two Patterns UWaveGestureLibraryAll Wine WordsSynonyms Worms WormsTwoClass synthetic control uwavegesturelibrary X uwavegesturelibrary Y uwavegesturelibrary Z wafer yoga
9 : a Differentiable Loss Function for Time-Series H. Barycenters: DTW loss (Eq. () withγ = ) achieved with init Dataset γ = γ =. γ =. γ =. Subgradient method mean 5words Adiac ArrowHead Beef BeetleFly BirdChicken CBF Car ChlorineConcentration CinC ECG torso Coffee Computers Cricket X Cricket Y Cricket Z DiatomSizeReduction DistalPhalanxOutlineAgeGroup DistalPhalanxOutlineCorrect DistalPhalanxTW ECG ECG ECGFiveDays Earthquakes ElectricDevices FISH FaceAll FaceFour FacesUCR FordA FordB Gun Point Ham HandOutlines Haptics Herring InlineSkate N/A N/A N/A ItalyPowerDemand MedicalImages MiddlePhalanxOutlineAgeGroup MiddlePhalanxOutlineCorrect MiddlePhalanxTW MoteStrain NonInvasiveFatalECG Thorax NonInvasiveFatalECG Thorax OSULeaf OliveOil PhalangesOutlinesCorrect Phoneme Plane ProximalPhalanxOutlineAgeGroup ProximalPhalanxOutlineCorrect ProximalPhalanxTW RefrigerationDevices ScreenType ShapeletSim ShapesAll SmallKitchenAppliances SonyAIBORobotSurface SonyAIBORobotSurfaceII StarLightCurves Strawberry SwedishLeaf Symbols ToeSegmentation ToeSegmentation Trace TwoLeadECG Two Patterns UWaveGestureLibraryAll Wine WordsSynonyms Worms WormsTwoClass synthetic control uwavegesturelibrary X uwavegesturelibrary Y uwavegesturelibrary Z wafer yoga
10 : a Differentiable Loss Function for Time-Series I.k-means clustering: DTW loss achieved (Eq. (5) withγ =, log-scaled) when using random initialization Dataset γ = γ =. γ =. γ =. Subgradient method mean 5words Adiac ArrowHead Beef BeetleFly BirdChicken CBF Car ChlorineConcentration CinC ECG torso Coffee Computers Cricket X Cricket Y Cricket Z DiatomSizeReduction DistalPhalanxOutlineAgeGroup DistalPhalanxOutlineCorrect DistalPhalanxTW ECG ECG ECGFiveDays Earthquakes ElectricDevices FISH FaceAll FaceFour FacesUCR FordA FordB Gun Point Ham MedicalImages MiddlePhalanxOutlineAgeGroup MiddlePhalanxOutlineCorrect MiddlePhalanxTW MoteStrain NonInvasiveFatalECG Thorax 7.78 N/A N/A N/A N/A N/A N/A ProximalPhalanxTW RefrigerationDevices ScreenType ShapeletSim ShapesAll SmallKitchenAppliances SonyAIBORobotSurface SonyAIBORobotSurfaceII StarLightCurves N/A N/A N/A N/A Trace TwoLeadECG Two Patterns UWaveGestureLibraryAll Wine WordsSynonyms Worms WormsTwoClass synthetic control uwavegesturelibrary X N/A N/A N/A N/A N/A
11 : a Differentiable Loss Function for Time-Series J.k-means clustering: DTW loss achieved (Eq. (5) withγ =, log-scaled) when using mean initialization Dataset γ = γ =. γ =. γ =. Subgradient method mean 5words Adiac ArrowHead Beef BeetleFly BirdChicken CBF Car ChlorineConcentration CinC ECG torso Coffee Computers Cricket X Cricket Y Cricket Z DiatomSizeReduction DistalPhalanxOutlineAgeGroup DistalPhalanxOutlineCorrect DistalPhalanxTW ECG ECG ECGFiveDays Earthquakes ElectricDevices FISH FaceAll FaceFour FacesUCR FordA FordB Gun Point Ham HandOutlines.7 N/A N/A N/A N/A N/A N/A MedicalImages MiddlePhalanxOutlineAgeGroup MiddlePhalanxOutlineCorrect MiddlePhalanxTW MoteStrain NonInvasiveFatalECG Thorax N/A N/A N/A N/A N/A N/A ProximalPhalanxTW RefrigerationDevices ScreenType ShapeletSim ShapesAll SmallKitchenAppliances SonyAIBORobotSurface SonyAIBORobotSurfaceII StarLightCurves N/A N/A N/A N/A Trace TwoLeadECG Two Patterns UWaveGestureLibraryAll Wine WordsSynonyms Worms WormsTwoClass synthetic control uwavegesturelibrary X N/A N/A N/A N/A N/A
12 : a Differentiable Loss Function for Time-Series K. Time-series prediction: DTW loss achieved when using random init Dataset loss γ = γ =. γ =. γ =. loss 5words Adiac ArrowHead Beef BeetleFly BirdChicken CBF Car ChlorineConcentration CinC ECG torso Coffee Computers Cricket X Cricket Y Cricket Z DiatomSizeReduction DistalPhalanxOutlineAgeGroup DistalPhalanxOutlineCorrect DistalPhalanxTW ECG ECG ECGFiveDays Earthquakes ElectricDevices FISH FaceAll FaceFour FacesUCR Gun Point Ham Haptics Herring InsectWingbeatSound ItalyPowerDemand LargeKitchenAppliances Lighting Lighting Meat MedicalImages MiddlePhalanxOutlineAgeGroup MiddlePhalanxOutlineCorrect MiddlePhalanxTW MoteStrain NonInvasiveFatalECG Thorax OSULeaf OliveOil PhalangesOutlinesCorrect Phoneme Plane ProximalPhalanxOutlineAgeGroup ProximalPhalanxOutlineCorrect ProximalPhalanxTW RefrigerationDevices ShapeletSim ShapesAll SonyAIBORobotSurface SonyAIBORobotSurfaceII Strawberry SwedishLeaf Symbols ToeSegmentation ToeSegmentation Trace TwoLeadECG Two Patterns UWaveGestureLibraryAll Wine WordsSynonyms Worms WormsTwoClass synthetic control uwavegesturelibrary X uwavegesturelibrary Y uwavegesturelibrary Z wafer yoga
13 : a Differentiable Loss Function for Time-Series L. Time-series prediction: DTW loss achieved when using init Dataset loss γ = γ =. γ =. γ =. loss 5words Adiac ArrowHead Beef BeetleFly BirdChicken CBF Car ChlorineConcentration CinC ECG torso Coffee Computers Cricket X Cricket Y Cricket Z DiatomSizeReduction DistalPhalanxOutlineAgeGroup DistalPhalanxOutlineCorrect DistalPhalanxTW ECG ECG ECGFiveDays Earthquakes ElectricDevices FISH FaceAll FaceFour FacesUCR Gun Point Ham Haptics Herring InsectWingbeatSound ItalyPowerDemand LargeKitchenAppliances Lighting Lighting Meat MedicalImages MiddlePhalanxOutlineAgeGroup MiddlePhalanxOutlineCorrect MiddlePhalanxTW MoteStrain NonInvasiveFatalECG Thorax OSULeaf OliveOil PhalangesOutlinesCorrect Phoneme Plane ProximalPhalanxOutlineAgeGroup ProximalPhalanxOutlineCorrect ProximalPhalanxTW RefrigerationDevices ShapeletSim ShapesAll SonyAIBORobotSurface SonyAIBORobotSurfaceII Strawberry SwedishLeaf Symbols ToeSegmentation ToeSegmentation Trace TwoLeadECG Two Patterns UWaveGestureLibraryAll Wine WordsSynonyms Worms WormsTwoClass synthetic control uwavegesturelibrary X uwavegesturelibrary Y uwavegesturelibrary Z wafer yoga
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