TY - GEN

T1 - Extended regression on manifolds estimation

AU - Kuleshov, Alexander

AU - Bernstein, Alexander

PY - 2016

Y1 - 2016

N2 - Let f(X) be unknown smooth function which maps p-dimensional manifold-valued inputs X, whose values lie on unknown Input manifold M of lower dimensionality q < p embedded in an ambient high-dimensional space Rp, to m-dimensional outputs. Regression on manifold problem is to estimate a triple (f(X), Jf(X), M), which includes Jacobian Jf of the mapping f, from given sample consisting of ‘input-output’ pairs. If some mapping h transforms Input manifold M to q-dimensional Feature space Yh = h(M) and satisfies certain conditions, initial estimating problem can be reduced to Regression on feature space problem consisting in estimating of triple (gf(y), Jg,f(y), Yh) in which unknown function gf(y) depends on low-dimensional features y = h(X) and satisfies the condition gf(h(X)) ≈ f(X), and Jg,f is its Jacobian. The paper considers such Extended problem and presents geometrically motivated method for estimating both triples from given sample.

AB - Let f(X) be unknown smooth function which maps p-dimensional manifold-valued inputs X, whose values lie on unknown Input manifold M of lower dimensionality q < p embedded in an ambient high-dimensional space Rp, to m-dimensional outputs. Regression on manifold problem is to estimate a triple (f(X), Jf(X), M), which includes Jacobian Jf of the mapping f, from given sample consisting of ‘input-output’ pairs. If some mapping h transforms Input manifold M to q-dimensional Feature space Yh = h(M) and satisfies certain conditions, initial estimating problem can be reduced to Regression on feature space problem consisting in estimating of triple (gf(y), Jg,f(y), Yh) in which unknown function gf(y) depends on low-dimensional features y = h(X) and satisfies the condition gf(h(X)) ≈ f(X), and Jg,f is its Jacobian. The paper considers such Extended problem and presents geometrically motivated method for estimating both triples from given sample.

KW - Input manifold estimation

KW - Jacobian estimation

KW - Regression on feature space

KW - Regression on manifolds

KW - Tangent bundle manifold learning

UR - http://www.scopus.com/inward/record.url?scp=84964018595&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-33395-3_15

DO - 10.1007/978-3-319-33395-3_15

M3 - Conference contribution

AN - SCOPUS:84964018595

SN - 9783319333946

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 208

EP - 228

BT - Conformal and Probabilistic Prediction with Applications - 5th International Symposium, COPA 2016, Proceedings

A2 - Vega, Jesus

A2 - Gammerman, Alexander

A2 - Luo, Zhiyuan

A2 - Vovk, Vladimir

PB - Springer Verlag

T2 - 5th International Symposium on Conformal and Probabilistic Prediction with Applications, COPA 2016

Y2 - 20 April 2016 through 22 April 2016

ER -