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532 Rise of the machinesis that the flow passing through x is a gradient ascent path moving towardshigher values of p. Unlike the paths defined by the gradient g which movetowards modes, the paths defined by G move towards ridges.The paths can be parameterized in many ways. One commonly used parameterizationis to use t ∈ [−∞, ∞] where large values of t correspond tohigher values of p. Inthiscaset = ∞ will correspond to a point on the ridge.In this parameterization we can express each integral curve in the flow as follows.A map π : R → R D is an integral curve with respect to the flow of Gifπ ′ (t) =G{π(t)} = L{π(t)}g{π(t)}. (44.10)Definition. The ridge R consists of the destinations of the integral curves:y ∈ R if lim t→∞ π(t) =y for some π satisfying (44.10).As mentioned above, the integral curves partition the space and for eachx/∈ R, thereisauniquepathπ x passing through x. The ridge points are zerosof the projected gradient: y ∈ R implies that G(y) =(0,...,0) ⊤ .Ozertemand Erdogmus (2011) derived an extension of the mean-shift algorithm, calledthe subspace constrained mean shift algorithm that finds ridges which can beapplied to the kernel density estimator. Our results can be summarized asfollows:1. Stability. We showed that if two functions are sufficiently close togetherthen their ridges are also close together (in Hausdorff distance).2. We constructed an estimator ̂R such thatH(R, ̂R) =O P( (log nn) 2)D+8(44.11)where H is the Hausdorff distance. Further, we showed that ̂R is topologicallysimilar to R. We also construct an estimator ̂R h for h>0 thatsatisfies( (log ) 1)H(R h , ̂R n2h )=O P , (44.12)nwhere R h is a smoothed version of R.3. Suppose the data are obtained by sampling points on a manifold andadding noise with small variance σ 2 .Weshowedthattheresultingdensityp has a ridge R σ such thatH(M,R σ )=O ( σ 2 log 3 (1/σ) ) (44.13)and R σ is topologically similar to M. Hence when the noise σ is small, theridge is close to M. Itthenfollowsthat( (log ) 2)H(M, ̂R) nD+8=O P + O ( σ 2 log 3 (1/σ) ) . (44.14)n