By Charles Hutton
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Extra resources for A Course of Mathematics. For the Use of Academies as Well as Private Tuition. Volume I (of Two)
Estimate/predict the output ζ from the noisy measurements y by: yi − vi ζˆi = xi if i ∈ Ia (estimation) if i ∈ Ie (interpolation/prediction) Examples of application of this idea are shown in Fig. 11, where it was used to establish target identity across occlusion, and in Fig. 12 where nonlinear embeddings were used first to map the data to a low order manifold where the rankminimization based interpolation was performed, followed by a remapping of the data to pixel space. Finally, Fig. 13 shows how a combination of dynamic interpolation and Hankel-rank based segmentation is able to detect occluded events.
In this context, noise can be readily handled by simply adding a new variable v such that the measured data 18 Sznaier et al. y = d + v, and a suitable noise description of the form v ∈ N , a convex, compact set. Finally, since rank minimization is NP-hard , using the convex relaxation proposed in  leads to the following algorithm: Algorithm 1: H ANKEL R ANK M INIMIZATION BASED I NTERPOLATION / P REDICTION Input at time k: Nh : Horizon length; Ia ⊆ [k − Nh , k]: set of indices of available measurements (with card(Ia ) ≥ n); Ie ⊆ [k − Nh , k + 1]: set of indices of data to be estimated; with Ia ∪ Ie = I; available data y , ∈ Ia ; set membership description of the measurement noise v ∈ N .
Subdifferentials in Theorem 2 are replaced by derivatives Φ (β −1 ) and Ψ (β) if Ψ and Φ are differentiable. This is the case when F (y) is strictly convex. Example 4 When solutions y¯ are in the exponential from (Example 3), one obtains U = x, y¯ = x eβx−Ψ (β) dy0 , and condition U = Ψ (β) gives Ψ (β) = ln eβ x(ω) dy0 (ω) Ω The above is the cumulant generating function of measure y0 . Potential Φ(β −1 ) = −β −1 Ψ (β) in this case is the free energy. 2 Information Trajectory of Optimal Learning 39 Fig.