Homework Answers
Yes, The height of the quadtree for S will be smaller than the height of the kd tree for S. In case of k-dimensions, each node can be considered as a grouping of k cosecutive level of a kd tree making the branching factor in order of 2^k. Comparision is done among all k dimensions at each node by making a partition into 2^k subdomains around a single k-dimensional point. Thus, making expected height of a quadtree less than the expected height of the kd tree by factor k.
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d) Let S be a set of two dimensional points. Can the height of the quadtree...
k-d tree Background One generalization of binary trees is the k-d tree, which stores k-dimensional data....
k-d tree Background One generalization of binary trees is the k-d tree, which stores k-dimensional data. Every internal node of a k-d tree indicates the dimension d and the value v in that dimension that it discriminates by. An internal node has exactly two children, containing data that is less-than-or-equal and data that is greater than v in dimension d. For example, if the node distinguishes on dimension 1, value 107, then the left child is for data with y...
Let A denote the set of all points (x, y) in the two-dimensional plane that satisfy the con strai...
Let A denote the set of all points (x, y) in the two-dimensional plane that satisfy the con straints: (a) 2 + y2 < 2, and (b)-> 0. Note that A is a semi-circular region. Consider a pair of random variables (X,Y) that is unifornly distributed over the region A. (a) Find the marginal pdf of X. (b) Find the marginal pdf of Y.
Let A denote the set of all points (x, y) in the two-dimensional plane that satisfy...
1) For the following set of two-dimensional points, draw a sketch of how they would be split into...
1) For the following set of two-dimensional points, draw a sketch of how they would be split into two clusters by K-means (when global minimum of SSE is achieved) and by Gaussian mixture model clustering. You can assume the density of points in the darker area is much higher than the density of points in the lighter area 2) Name one other clustering method that might be able to accurately capture the two clusters.
1) For the following set of...
Exercise 25. Let , be an orthonormal basis of a two-dimensional subspace S of R" and...
Exercise 25. Let , be an orthonormal basis of a two-dimensional subspace S of R" and A xyT + (i) Show that x+y and x -y are eigenvectors of A. What are their corresponding eigenvalues? (ii) Show that 0 is an eigenvalue of R" with n - 2 linearly independent eigenvectors. (iii) Explain why A is diagonalizable.
Exercise 25. Let , be an orthonormal basis of a two-dimensional subspace S of R" and A xyT + (i) Show that x+y...
Exercise 20 Let fr,y} be an orthonormal basis of a two-dimensional subspace S of R" and...
Exercise 20 Let fr,y} be an orthonormal basis of a two-dimensional subspace S of R" and T A= xx SL (i) Show that N(A (ii) Show that the rank of A is 2 (iii Show that x and y are eigenvectors of A for an eigenvalue X of A. What is A?
Exercise 20 Let fr,y} be an orthonormal basis of a two-dimensional subspace S of R" and T A= xx SL (i) Show that N(A (ii) Show that the...
3) Let S be a set with an associative binary operation :SxS->S. Let e, be a...
3) Let S be a set with an associative binary operation :SxS->S. Let e, be a left identity of S (i.e., e, *ssVse S), and let eg be a right identity of S (i.e., a) Prove that e-e b) Also prove that S can have at most one 2-sided identity.
6) Let S be a subset of an m-dimensional vector space and suppose S contains fewer...
6) Let S be a subset of an m-dimensional vector space and suppose S contains fewer than m'vectors. Explain why s cannot span V. (itiut: Assume S does pan, there is a subset of that is a but. which is a contradictor) then using 2 freed more than for words here
11. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g}...
11. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. Find: A ∪ B 12. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. Find: b. A ∩ B 13. Let the universal set be the set U = {a,b,c,d,e,f,g} and let A = {a,c,e,g} and B = {d, e, f, g}. Find: AC...
2. (5 points) Let {sn}nen be a sequence. Let S be the set of subsequential limits...
2. (5 points) Let {sn}nen be a sequence. Let S be the set of subsequential limits of {Sn}nen, that is, x E S if and only if 3{Sn}ken subsequence of {Sn}nen such that limky Sny = x. Use the previous problem to show that inf S = lim inf sns sup S = lim sup sn.
Q7 8 Points Let V, W, and U be three finite dimensional vector spaces over R...
Q7 8 Points Let V, W, and U be three finite dimensional vector spaces over R and T:V + Wand S : W → U be two linear transformations. Q7.1 4 Points Show that null(So T) < null(T) + null(S) Please select file(s) Select file(s) Save Answer Q7.2 4 Points Show that rank(S • T) > rank(T) + rank(S) – dim(W) (Hint: Use part (1) at some point)
Let A denote the set of all points (x, y) in the two-dimensional plane that satisfy the con straints: (a) 2 + y2 < 2, and (b)-> 0. Note that A is a semi-circular region. Consider a pair of random variables (X,Y) that is unifornly distributed over the region A. (a) Find the marginal pdf of X. (b) Find the marginal pdf of Y.
Let A denote the set of all points (x, y) in the two-dimensional plane that satisfy...
1) For the following set of two-dimensional points, draw a sketch of how they would be split into two clusters by K-means (when global minimum of SSE is achieved) and by Gaussian mixture model clustering. You can assume the density of points in the darker area is much higher than the density of points in the lighter area 2) Name one other clustering method that might be able to accurately capture the two clusters.
1) For the following set of...
Exercise 25. Let , be an orthonormal basis of a two-dimensional subspace S of R" and A xyT + (i) Show that x+y and x -y are eigenvectors of A. What are their corresponding eigenvalues? (ii) Show that 0 is an eigenvalue of R" with n - 2 linearly independent eigenvectors. (iii) Explain why A is diagonalizable.
Exercise 25. Let , be an orthonormal basis of a two-dimensional subspace S of R" and A xyT + (i) Show that x+y...
Exercise 20 Let fr,y} be an orthonormal basis of a two-dimensional subspace S of R" and T A= xx SL (i) Show that N(A (ii) Show that the rank of A is 2 (iii Show that x and y are eigenvectors of A for an eigenvalue X of A. What is A?
Exercise 20 Let fr,y} be an orthonormal basis of a two-dimensional subspace S of R" and T A= xx SL (i) Show that N(A (ii) Show that the...
3) Let S be a set with an associative binary operation :SxS->S. Let e, be a left identity of S (i.e., e, *ssVse S), and let eg be a right identity of S (i.e., a) Prove that e-e b) Also prove that S can have at most one 2-sided identity.
6) Let S be a subset of an m-dimensional vector space and suppose S contains fewer than m'vectors. Explain why s cannot span V. (itiut: Assume S does pan, there is a subset of that is a but. which is a contradictor) then using 2 freed more than for words here
2. (5 points) Let {sn}nen be a sequence. Let S be the set of subsequential limits of {Sn}nen, that is, x E S if and only if 3{Sn}ken subsequence of {Sn}nen such that limky Sny = x. Use the previous problem to show that inf S = lim inf sns sup S = lim sup sn.
Q7 8 Points Let V, W, and U be three finite dimensional vector spaces over R and T:V + Wand S : W → U be two linear transformations. Q7.1 4 Points Show that null(So T) < null(T) + null(S) Please select file(s) Select file(s) Save Answer Q7.2 4 Points Show that rank(S • T) > rank(T) + rank(S) – dim(W) (Hint: Use part (1) at some point)
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