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Non-parametric Online Change Point Detection on Riemannian Manifolds
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Proposes a non-parametric online change point detection method for Riemannian manifold-valued streams that compares two R-SGD estimates of the generalized Karcher mean with different constant step sizes, with theoretical false-alarm and detection bounds.
Problem
Online change point detection in streaming data lying on Riemannian manifolds, where Euclidean non-parametric CPD methods do not account for manifold geometry and prior theory/online operation is lacking.
Approach
Monitor the distance between two stochastic Riemannian gradient descent estimates of the generalized Karcher mean, one with a small step size (long memory) and one with a large step size (fast adaptation), extending the NEWMA idea from Euclidean spaces to manifolds, with non-asymptotic convergence results for constant-step R-SGD.</120>
Key results
- On synthetic SPD and Grassmann data, the proposed method achieves significantly lower detection delay than Scan-B, NEWMA, F-CPD and NODE at a fixed ARL (averaged over 10^4 Monte Carlo runs).
- On voice activity detection, the proposed method shows higher detection rate at a given false alarm rate than Scan-B and NEWMA, except at very small ARL where Scan-B has lower MDD.
- On HDM05 skeleton action recognition, the method achieves a significantly higher detection rate than Scan-B and NEWMA, which had very similar ROCs.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use Riemannian geometry and recursive online estimation instead of Euclidean-space CPD methods | Proposed method vs. Scan-B, NEWMA, and NODE · Synthetic SPD matrices | mean detection delay at fixed average run length | Helped |
| Use Riemannian geometry and recursive online estimation instead of Euclidean-space CPD methods | Proposed method vs. Scan-B, NEWMA, and NODE · Synthetic Grassmann data | mean detection delay at fixed average run length | Helped |
| Use the proposed manifold-based online CPD method instead of Scan-B or NEWMA | Voice activity detection · Voice activity detection (real audio) | detection rate at a given false alarm rate | Helped |
| Use the proposed manifold-based online CPD method instead of Scan-B or NEWMA | Voice activity detection · Voice activity detection (real audio) | mean detection delay versus average run length | Mixed |
| Use the proposed manifold-based online CPD method instead of Scan-B or NEWMA | Skeleton-based action recognition · HDM05 skeleton-based action recognition | detection rate | Helped |
| Use the proposed manifold-based online CPD method instead of NEWMA | Skeleton-based action recognition · HDM05 skeleton-based action recognition | mean detection delay | Mixed |
Surprising findings
- Performance on the Grassmann manifold, which is not Hadamard, remains empirically good despite lacking the theoretical assumptions.
Limitations
- Requires the sample count between change points to be large enough for the slow estimator to converge.
- Theoretical analysis assumes Hadamard manifolds and additional conditions such as strong convexity and smoothness, and convergence slows with higher curvature.
- Non-convexity on complex manifolds may complicate the cost function.
Experimental setup
Datasets: TIMIT QUT-NOISE HDM05
Benchmarks: synthetic Wishart SPD data synthetic matrix Gaussian Grassmann data TIMIT with QUT-NOISE voice activity detection HDM05 skeleton action recognition
Metrics: ROC curve (detection rate vs false alarm rate) average run length mean detection delay
Baselines: Scan-B NEWMA F-CPD NODE
Builds on: NEWMA R-SGD (Bonnabel 2013) Karcher mean Zhang & Sra 2016 geodesic convex optimization
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