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Linear Separability in Non-Euclidean Geometry

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Linear Separability in Non-Euclidean Geometry Linear Separability George M. Georgiou Linear Separability in Non-Euclidean Outline Geometry The problem Separating hyperplane Euclidean George M. Georgiou, Ph.D. Geometry Non-Euclidean Geometry Computer Science Department California State University, San Bernardino Three solutions Conclusion March 3, 2006 [email protected] 1 : 29 The problem Linear Separability Find a geodesic that separates two given sets of points on George M. Georgiou the Poincare´ disk non-Euclidean geometry model. Outline The problem Linear Separability Geometrically Separating hyperplane Euclidean Geometry Non-Euclidean Geometry Three solutions Conclusion 2 : 29 The problem Linear Separability Find a geodesic that separates two given sets of points on George M. Georgiou the Poincare´ disk non-Euclidean geometry model. Outline The problem Linear Separability Geometrically Separating hyperplane Euclidean Geometry Non-Euclidean Geometry Three solutions Conclusion 2 : 29 Linear Separability in Machine Learning Linear Separability George M. Georgiou Machine Learning is branch of AI, which includes areas Outline such as pattern recognition and artificial neural networks. The problem n Linear Separability Each point X = (x1, x2,..., xn) ∈ (vector or object) Geometrically R Separating belongs to two class C0 and C1 with labels -1 and 1, hyperplane respectively. Euclidean Geometry The two classes are linearly separable if there exists n+1 Non-Euclidean W = (w0, w1, w2,..., wn) ∈ such that Geometry R Three solutions w0 + w1x1 + w2x2 + ... + wnxn 0, if X ∈ C0 (1) Conclusion > w0 + w1x1 + w2x2 + ... + wnxn< 0, if X ∈ C1 (2) 3 : 29 Geometrically Linear Separability George M. Georgiou Outline The problem Linear Separability Geometrically Separating hyperplane Euclidean Geometry Non-Euclidean Geometry Three solutions Conclusion 4 : 29 “Linearizing” the circle Linear Separability George M. Georgiou Outline The problem Linear Separability Geometrically Separating hyperplane Euclidean Geometry Non-Euclidean Geometry (x, y) −→ (x, y, x2 + y 2) (3) Three solutions Conclusion Since any circle can be written as 2 2 w0 + w1x + w2y + w3(x + y ) = 0 (4) 5 : 29 Separating hyperplane Linear Separability George M. Georgiou Outline The problem Linear Separability Geometrically w0 + w1x1 + w2x2 + ... + wnxn= 0 Separating n = 2, 2-D: line hyperplane Euclidean n = 3, 3-D: plane Geometry n > 3, higher dimensions: hyperplane Non-Euclidean Geometry Three solutions Conclusion 6 : 29 Finding the separating hyperplane Linear Separability George M. Georgiou Outline The problem A number of ways Separating hyperplane Use the perceptron algorithm Perceptron Linear inequalities Solve a system of linear inequalities Euclidean Geometry Use Support Vector Machine (SVM) method. It maximizes Non-Euclidean Geometry separation between the two classes. Three solutions Conclusion 7 : 29 Finding the separating hyperplane Linear Separability George M. Georgiou Outline The problem A number of ways Separating hyperplane Use the perceptron algorithm Perceptron Linear inequalities Solve a system of linear inequalities Euclidean Geometry Use Support Vector Machine (SVM) method. It maximizes Non-Euclidean Geometry separation between the two classes. Three solutions Conclusion 7 : 29 Finding the separating hyperplane Linear Separability George M. Georgiou Outline The problem A number of ways Separating hyperplane Use the perceptron algorithm Perceptron Linear inequalities Solve a system of linear inequalities Euclidean Geometry Use Support Vector Machine (SVM) method. It maximizes Non-Euclidean Geometry separation between the two classes. Three solutions Conclusion 7 : 29 The perceptron (Rosenblatt, 1958) Linear w0 Separability 1 George M. x1 w1 Georgiou f(y) w2 x2 Outline w3 The problem x3 Separating w4 hyperplane Perceptron x4 Linear inequalities i=n Euclidean Geometry 1, if y 0 y = w x (6) f (y) = > (5) i i Non-Euclidean −1, if y < 0 i=0 Geometry X Three solutions Perceptron Algorithm Conclusion 1 Start with W = 0. 2 Present input vector X and compute error ε = d − f (y). 3 Update W using ∆W = α ε X 4 Pick another input vector X and goto to step 2. 8 : 29 Solving linear inequalities Linear Separability George M. Georgiou Outline The problem Separating Can use linear programming, e.g. simplex method, hyperplane Karmakar, and variants. Perceptron Linear inequalities Polynomial complexity Euclidean Geometry In fact, when dimension is fixed, e.g. d=2,3 as in our case, Non-Euclidean Geometry complexity is linear in the number of input vectors. Three solutions Conclusion 9 : 29 Euclidean Geometry Linear Separability George M. Axioms of Euclid (ca. 400 BC) Georgiou I. Two points determine a unique line. Outline The problem II. A line is of infinite length. Separating III. A circle may be described with any center at any distance hyperplane from the center. Euclidean Geometry IV. All right angles are equal. Non-Euclidean Geometry V. If a straight line meet two other straight lines, so as to Three solutions make the two interior angles on one side of it together less Conclusion than two right angles, the other straight lines will meet if produced on that side on which the angles are less than two right angles. (from H.S.M. Coxeter, Non-Euclidean Geometry) 10 : 29 Non-Euclidean Geometry Linear Separability George M. Georgiou Euclid’s Elements considered to be the most successful textbook of all times. Outline The problem Nicolai Lobachevsky (1829) and Janos´ Bolyai (1832) first Separating to publish geometries without the Euclid’s axiom V. hyperplane Euclidean Gauss feared the “howling of the Boetians.” Geometry Non-Euclidean Geometry Types on Non-Euclidean Geometry Poincare´ Disk Lobachevskian (hyperbolic geometry) Geometry: from a Three solutions point not on a line, an infinite number of parallel lines can Conclusion be drawn. Riemannian (elliptic) geometry: no parallel lines can be drawn. 11 : 29 Poincare´ disk Linear Separability George M. Georgiou Outline The problem Separating hyperplane Euclidean Geometry Non-Euclidean bM Geometry Poincare´ Disk bN Three solutions Conclusion 12 : 29 Poincare´ disk Linear Separability George M. Georgiou Outline The problem Separating hyperplane Euclidean b P Geometry Non-Euclidean bM Geometry Poincare´ Disk bN Three solutions Conclusion 12 : 29 Distance function Linear Separability George M. Georgiou The distance between points z and z , taken as complex Outline 1 2 The problem numbers, is Separating D(z1, z2) = | log(z1, z2; µ, λ)|, (7) hyperplane Euclidean where z1 and z2 are points on the unit disk, and µ and λ are Geometry the intersections of the geodesic defined by z1 and z2 with the Non-Euclidean Geometry unit circle. The cross ratio (z1, z2; µ, λ), is defined as Poincare´ Disk Three solutions z − µ.z − µ (z , z ; µ, λ) = 1 2 (8) Conclusion 1 2 z1 − λ z2 − λ 13 : 29 Art by M.C. Esher (1878–1972) Limit Circle III Linear Separability George M. Georgiou Outline The problem Separating hyperplane Euclidean Geometry Non-Euclidean Geometry Poincare´ Disk Three solutions Conclusion 14 : 29 The problem Linear Separability Find a geodesic that separates two given sets of points on George M. Georgiou the Poincare´ disk non-Euclidean geometry model. Outline The problem Separating hyperplane Euclidean Geometry Non-Euclidean Geometry Poincare´ Disk Three solutions Conclusion 16 : 29 The problem Linear Separability Find a geodesic that separates two given sets of points on George M. Georgiou the Poincare´ disk non-Euclidean geometry model. Outline The problem Separating hyperplane Euclidean Geometry Non-Euclidean Geometry Poincare´ Disk Three solutions Conclusion 16 : 29 Three solutions Linear Separability George M. Georgiou Outline 1 Map points from Poincare´ disk to the Klein disk, find The problem separating line, which maps to a desired geodesic on the Separating Poincare´ disk. hyperplane 2 Euclidean Map points from Poincare´ disk to the Hyperboloid model, Geometry find separating plane, which maps to a desired geodesic Non-Euclidean Geometry on the Poincare´ disk. Three solutions 3 Map points from Poincare´ disk to circular paraboloid, find Klein disk Hyperboloid model separating plane, which maps to a desired geodesic on Circular Paraboloid Conclusion the Poincare´ disk. (This is a novel map.) 17 : 29 Three solutions Linear Separability George M. Georgiou Outline 1 Map points from Poincare´ disk to the Klein disk, find The problem separating line, which maps to a desired geodesic on the Separating Poincare´ disk. hyperplane 2 Euclidean Map points from Poincare´ disk to the Hyperboloid model, Geometry find separating plane, which maps to a desired geodesic Non-Euclidean Geometry on the Poincare´ disk. Three solutions 3 Map points from Poincare´ disk to circular paraboloid, find Klein disk Hyperboloid model separating plane, which maps to a desired geodesic on Circular Paraboloid Conclusion the Poincare´ disk. (This is a novel map.) 17 : 29 Three solutions Linear Separability George M. Georgiou Outline 1 Map points from Poincare´ disk to the Klein disk, find The problem separating line, which maps to a desired geodesic on the Separating Poincare´ disk. hyperplane 2 Euclidean Map points from Poincare´ disk to the Hyperboloid model, Geometry find separating plane, which maps to a desired geodesic Non-Euclidean Geometry on the Poincare´ disk. Three solutions 3 Map points from Poincare´ disk to circular paraboloid, find Klein disk Hyperboloid model separating plane, which maps to a desired geodesic on Circular Paraboloid Conclusion the Poincare´ disk. (This is a novel map.) 17 : 29 The Klein
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