machine learning - What is the difference between hyperplane and plane ? and why is hyperplane represented using eqn w^T+b=0 ? -

machine learning - What is the difference between hyperplane and plane ? and why is hyperplane represented using eqn w^T+b=0 ? -

i need brief thought svm.so guys plz help create me understand representation of hyperplane ? , thought of kernel ?

while first part of @jeffheaton 's reply nice, rest not answet op question, farther details:

formula

why hyperplane equationw^tx+b=0 ? first, have aware w^tx=<w,x> x. projects x (vector starting @ (0,0)) @ w (which vector starting in (0,0), result, can either positive number (angle between x , w less 90 degrees), equal 0 (they perpendicular) or negative (angle bigger 90 degrees). can see, equals 0 iff these 2 objects perpendicular, thing left distance origin (0,0) done adding constant b. geometrical point of view w called "normal hyperplane", - vector perpendicular hyperplane. so, if calculate <w,x> , 0, x perpendicular w, perpendicular hyperplane, x lies in hyperplane.

kernel

kernel, nil more scalar product <w,x> written in previous formula. reason writing k(x,y) instead, assumes, have "magical"mapping different space phi. in other words, if have function phi rearanges points in way, easier classify, can train linear svm on phi(x), l instead of x,l (where l right labels). problem is, hard find phi. in practise, take bir random or arbitrary phi, maps points higher dimension. known mathematical fact, in higher dimension, points easier separate. in particular, if have n points x_1, ..., x_n, can select such phi, phi(x_i)=[0 0 0 ... 1 ... 0], 1 appears on ith position. unfortunately suchphi(x) expensive calculate, , utilize kernel functions instead, defined k(x,y)=<phi(x), phi(y)>. not have know explicit phi value, instead, need know scalar product between images of points through phi. , kernels do, denote scalar products in differenct spaces. in particular, rbf kernel maps each point ... function (in fact, gaussian distribution). phi(x) has infinite dimension , cannot efficiently calculated, scalar product between 2 functions integral of multiplication, quite easy object.

machine-learning svm computational-geometry