Predictive Online Optimisation with Applications to Optical Flow

Abstract

Online optimisation revolves around new data being introduced into a problem while it is still being solved; think of deep learning as more training samples become available. We adapt the idea to dynamic inverse problems such as video processing with optical flow. We introduce a corresponding predictive online primal-dual proximal splitting method. The video frames now exactly correspond to the algorithm iterations. A user-prescribed predictor describes the evolution of the primal variable. To prove convergence we need a predictor for the dual variable based on (proximal) gradient flow. This affects the model that the method asymptotically minimises. We show that for inverse problems the effect is, essentially, to construct a new dynamic regulariser based on infimal convolution of the static regularisers with the temporal coupling. We finish by demonstrating excellent real-time performance of our method in computational image stabilisation and convergence in terms of regularisation theory.

Local Strong Convexity

We establish local strong convexity of the indicator function of the ball. This has been shown in [1] to be equivalent to the strong metric subregularity of the subdifferential. For related characterisations, see also [33] and regarding total variation [22, appendix].

Lemma A.1

With $F: X \rightarrow \mathbb {R}$, $F=\delta _{{{\,\mathrm{cl}\,}}B(0, \alpha )}$ on a Hilbert space X, suppose $x \in \partial B(0, \alpha )$ and $0 \ne x^* \in \partial F(x)$. Then,

$$\begin{aligned} F(x')-F(x) \ge \langle x^*,x'-x\rangle + \frac{\gamma }{2}\Vert x'-x\Vert ^2 \quad (x' \in U_x) \end{aligned}$$

for

$$\begin{aligned} U_x = {\left\{ \begin{array}{ll} X, &{} 0 \le \gamma \alpha \le \Vert x^*\Vert , \\ {[}{{\,\mathrm{cl}\,}}B(0,\alpha )]^c \cup {{\,\mathrm{cl}\,}}B(x, \alpha ), &{} \alpha \gamma > \Vert x^*\Vert . \end{array}\right. } \end{aligned}$$

Proof

Observe that $x^*=\lambda x$ for $\lambda :=\Vert x^*\Vert /\alpha$. If $x' \not \in {{\,\mathrm{cl}\,}}B(0, \alpha )$, there is nothing to prove. So take $x' \in {{\,\mathrm{cl}\,}}B(0, \alpha )$. Then, we need $0 \ge \lambda \langle x,x'-x\rangle + \frac{\gamma }{2}\Vert x'-x\Vert ^2.$ Since $\Vert x\Vert =\alpha$, this says

$$\begin{aligned} \left( \lambda -\frac{\gamma }{2}\right) \alpha ^2 \ge \frac{\gamma }{2}\Vert x'\Vert ^2 + \left( \lambda -\gamma \right) \langle x,x'\rangle . \end{aligned}$$

(A.1)

Suppose $\gamma \le \lambda$, which is the first case of $U_x$. Then (A.1) is seen to hold by application of Young’s inequality on the inner product term, followed by $\Vert x'\Vert \le \alpha$.

If on the other hand, $\gamma > \lambda$, which is the second case of $U_x$, we take $x' \in {{\,\mathrm{cl}\,}}B(x, \alpha ) \cap {{\,\mathrm{cl}\,}}B(0,\alpha )$. This implies $\langle x',x\rangle \ge \tfrac{1}{2}\Vert x'\Vert ^2.$ Since $\lambda -\gamma <0$, this and $\Vert x'\Vert \le \alpha$ prove (A.1). $\square$