- 2 From Distances To Embeddings Your Friend From Overseas Is Visiting You And Asks You The Geographical Locations Of Popu 1 (48.21 KiB) Viewed 24 times
2 From distances to embeddings Your friend from overseas is visiting you and asks you the geographical locations of popular US cities on a map. Not having access to a US map, you realize that you cannot provide your friend accurate information. You recall that you have access to the relative distances between nine popular US cities, given by the following distance matrix D: Distances (D) | BOS NYC DC MIA CHI SEA SF LA DEN BOS 0 206 4291504 963 2976 3095 2979 1949 NYC 206 0 233 1308 802 2815 2934 2786 1771 DC 429 233 0 1075 671 2684 2799 2631 1616 MIA 1504 1308 1075 0 1329 3273 3053 2687 2037 CHI 963 802 671 1329 0 2013 2142 2054 996 SEA 2976 2815 2684 3273 2013 0 808 1131 1307 SF 3095 2934 2799 3053 2142 808 0 379 1235 LA 2979 2786 2631 2687 2054 1131 379 0 1059 DEN 1949 1771 1616 2037 996 1307 1235 1059 0 0 Being a machine learning student, you believe that it may be possible to infer the locations of these cities from the distance data. To find an embedding of these nine cities on a two dimensional map, you decide to solve it as an optimization problem as follows.
(1) Let Z4 := D4+1() (i.e., Z4 denotes the normalization constant for the weighted distribu- tion D++1). Show that 1 1 Dr+1(i) m [1Zexp(-4:9(;). (ii) Show that error of the aggregate classifier g is upper bounded by the product of Z:: err(9) < Π. Ζ. (hint: use the fact that 0-1 loss is upper bounded by exponential loss)