Today. Cuckoo hashing. Today. Cuckoo hashing. - PowerPoint PPT Presentation

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions.

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k y i

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k y i = � v i | x �

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k y i = � v i | x � = � Uv i | Ux �

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k y i = � v i | x � = � Uv i | Ux � = � e i | Ux �

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k y i = � v i | x � = � Uv i | Ux � = � e i | Ux � = � e i | z �

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k y i = � v i | x � = � Uv i | Ux � = � e i | Ux � = � e i | z � Inverse of U maps e i to random vector v i

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k y i = � v i | x � = � Uv i | Ux � = � e i | Ux � = � e i | z � Inverse of U maps e i to random vector v i

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k y i = � v i | x � = � Uv i | Ux � = � e i | Ux � = � e i | z � Inverse of U maps e i to random vector v i z = Ux is uniformly distributed on d sphere for unit x ∈ R d .

Projections. Project x into subspace spanned by v 1 , v 2 , ··· , v k . y 1 = x · v 1 , y 2 = x · , v 2 , ··· , y k = x · v k Projection: ( y 1 ,..., y k ) . Have: Arbitrary vector, random k -dimensional subspace. View As: Random vector, standard basis for k dimensions. Orthogonal U - rotates v 1 ,..., v k onto e 1 ,..., e k y i = � v i | x � = � Uv i | Ux � = � e i | Ux � = � e i | z � Inverse of U maps e i to random vector v i z = Ux is uniformly distributed on d sphere for unit x ∈ R d . y i is i th coordinate of random vector z .

Expected value of y i . Random projection: first k coordinates of random unit vector, z i .

Expected value of y i . Random projection: first k coordinates of random unit vector, z i . E [ ∑ i ∈ [ d ] z 2 i ] = 1.

Expected value of y i . Random projection: first k coordinates of random unit vector, z i . E [ ∑ i ∈ [ d ] z 2 i ] = 1. Linearity of Expectation.

Expected value of y i . Random projection: first k coordinates of random unit vector, z i . E [ ∑ i ∈ [ d ] z 2 i ] = 1. Linearity of Expectation. By symmetry, each z i is identically distributed.

Expected value of y i . Random projection: first k coordinates of random unit vector, z i . E [ ∑ i ∈ [ d ] z 2 i ] = 1. Linearity of Expectation. By symmetry, each z i is identically distributed. i ] = k E [ ∑ i ∈ [ k ] z 2 d .

Expected value of y i . Random projection: first k coordinates of random unit vector, z i . E [ ∑ i ∈ [ d ] z 2 i ] = 1. Linearity of Expectation. By symmetry, each z i is identically distributed. i ] = k E [ ∑ i ∈ [ k ] z 2 d . Linearity of Expectation.

Expected value of y i . Random projection: first k coordinates of random unit vector, z i . E [ ∑ i ∈ [ d ] z 2 i ] = 1. Linearity of Expectation. By symmetry, each z i is identically distributed. i ] = k E [ ∑ i ∈ [ k ] z 2 d . Linearity of Expectation. � k Expected length is d .

Expected value of y i . Random projection: first k coordinates of random unit vector, z i . E [ ∑ i ∈ [ d ] z 2 i ] = 1. Linearity of Expectation. By symmetry, each z i is identically distributed. i ] = k E [ ∑ i ∈ [ k ] z 2 d . Linearity of Expectation. � k Expected length is d . Johnson-Lindenstrass: close to expectation.

Expected value of y i . Random projection: first k coordinates of random unit vector, z i . E [ ∑ i ∈ [ d ] z 2 i ] = 1. Linearity of Expectation. By symmetry, each z i is identically distributed. i ] = k E [ ∑ i ∈ [ k ] z 2 d . Linearity of Expectation. � k Expected length is d . Johnson-Lindenstrass: close to expectation. k is large enough →

Expected value of y i . Random projection: first k coordinates of random unit vector, z i . E [ ∑ i ∈ [ d ] z 2 i ] = 1. Linearity of Expectation. By symmetry, each z i is identically distributed. i ] = k E [ ∑ i ∈ [ k ] z 2 d . Linearity of Expectation. � k Expected length is d . Johnson-Lindenstrass: close to expectation. k is large enough → � k ≈ ( 1 ± ε ) d with decent probability.

Concentration Bounds. z is uniformly random unit vector.

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d .

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 .

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . ∆

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if ∆

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if z ≥ ∆ from equator of sphere. ∆

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if z ≥ ∆ from equator of sphere. Point on “ ∆ -spherical cap”. ∆

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if z ≥ ∆ from equator of sphere. Point on “ ∆ -spherical cap”. Area of caps ∆

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if z ≥ ∆ from equator of sphere. Point on “ ∆ -spherical cap”. Area of caps √ ∆ ≤ S.A. of sphere of radius 1 − ∆ 2

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if z ≥ ∆ from equator of sphere. Point on “ ∆ -spherical cap”. Area of caps √ ∆ ≤ S.A. of sphere of radius 1 − ∆ 2 ∝ r d = 1 − ∆ 2 � d / 2 �

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if z ≥ ∆ from equator of sphere. Point on “ ∆ -spherical cap”. Area of caps √ ∆ ≤ S.A. of sphere of radius 1 − ∆ 2 ∝ r d = 1 − ∆ 2 � d / 2 � � d / 2 � 1 − t 2 ∝ d

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if z ≥ ∆ from equator of sphere. Point on “ ∆ -spherical cap”. Area of caps √ ∆ ≤ S.A. of sphere of radius 1 − ∆ 2 ∝ r d = 1 − ∆ 2 � d / 2 � � d / 2 − t 2 � 1 − t 2 ∝ ≈ e 2 d

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if z ≥ ∆ from equator of sphere. Point on “ ∆ -spherical cap”. Area of caps √ ∆ ≤ S.A. of sphere of radius 1 − ∆ 2 ∝ r d = 1 − ∆ 2 � d / 2 � � d / 2 − t 2 � 1 − t 2 ∝ ≈ e 2 d Constant of ∝ is unit sphere area.

Concentration Bounds. z is uniformly random unit vector. Random point on the unit sphere. E [ ∑ i ∈ [ k ] z 2 i ] = k d . d ] ≤ e − t 2 / 2 t Claim: Pr [ | z 1 | > √ Sphere view: surface “far” from equator defined by e 1 . | z 1 | ≥ ∆ if z ≥ ∆ from equator of sphere. Point on “ ∆ -spherical cap”. Area of caps √ ∆ ≤ S.A. of sphere of radius 1 − ∆ 2 ∝ r d = 1 − ∆ 2 � d / 2 � � d / 2 − t 2 � 1 − t 2 ∝ ≈ e 2 d Constant of ∝ is unit sphere area. Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small.

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small.

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. Total Length?

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k .

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d �

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 .

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k − � � d � �

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε d ] � � d �

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � d �

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d �

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d � Johnson-Lindenstraus: For n points, x 1 ,..., x n , all distances � k preserved to within 1 ± ε under d -scaled projection above.

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d � Johnson-Lindenstraus: For n points, x 1 ,..., x n , all distances � k preserved to within 1 ± ε under d -scaled projection above. View one pair x i − x j as vector.

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d � Johnson-Lindenstraus: For n points, x 1 ,..., x n , all distances � k preserved to within 1 ± ε under d -scaled projection above. View one pair x i − x j as vector. Scale to unit.

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d � Johnson-Lindenstraus: For n points, x 1 ,..., x n , all distances � k preserved to within 1 ± ε under d -scaled projection above. View one pair x i − x j as vector. Scale to unit. Projection fails to preserve | x i − x j |

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d � Johnson-Lindenstraus: For n points, x 1 ,..., x n , all distances � k preserved to within 1 ± ε under d -scaled projection above. View one pair x i − x j as vector. Scale to unit. Projection fails to preserve | x i − x j | with probability ≤ 1 n c

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d � Johnson-Lindenstraus: For n points, x 1 ,..., x n , all distances � k preserved to within 1 ± ε under d -scaled projection above. View one pair x i − x j as vector. Scale to unit. Projection fails to preserve | x i − x j | with probability ≤ 1 n c Scaled vector length also preserved.

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d � Johnson-Lindenstraus: For n points, x 1 ,..., x n , all distances � k preserved to within 1 ± ε under d -scaled projection above. View one pair x i − x j as vector. Scale to unit. Projection fails to preserve | x i − x j | with probability ≤ 1 n c Scaled vector length also preserved. ≤ n 2 pairs

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d � Johnson-Lindenstraus: For n points, x 1 ,..., x n , all distances � k preserved to within 1 ± ε under d -scaled projection above. View one pair x i − x j as vector. Scale to unit. Projection fails to preserve | x i − x j | with probability ≤ 1 n c Scaled vector length also preserved. ≤ n 2 pairs plus union bound

Many coordinates. Argued Pr [ any z 2 i > ( 2log d ) E [ z 2 i ]] is small. � z 2 1 + z 2 2 + ··· z 2 Total Length? z = k . � � � � � > t ] ≤ e − t 2 d / 2 � ( z 2 1 + z 2 2 + ··· + z 2 k � Pr [ k ) − � � d � � d , k = c log n k Substituting t = ε ε 2 . � � � � � d ] ≤ e − ε 2 k = e − c log n = 1 � z 2 1 + z 2 2 + ··· + z 2 k � k Pr [ k − � > ε � � n c d � Johnson-Lindenstraus: For n points, x 1 ,..., x n , all distances � k preserved to within 1 ± ε under d -scaled projection above. View one pair x i − x j as vector. Scale to unit. Projection fails to preserve | x i − x j | with probability ≤ 1 n c Scaled vector length also preserved. ≤ n 2 pairs plus union bound 1 → prob any pair fails to be preserved with ≤ n c − 2 .

Locality Preserving Hashing Find nearby points in high dimensional space.

Locality Preserving Hashing Find nearby points in high dimensional space. Points could be images!

Locality Preserving Hashing Find nearby points in high dimensional space. Points could be images! Hash function h ( · ) s.t. h ( x i ) = h ( x j ) if d ( x i , x j ) ≤ δ .

Locality Preserving Hashing Find nearby points in high dimensional space. Points could be images! Hash function h ( · ) s.t. h ( x i ) = h ( x j ) if d ( x i , x j ) ≤ δ . √ Low dimensions: grid cells give d -approximation.

Locality Preserving Hashing Find nearby points in high dimensional space. Points could be images! Hash function h ( · ) s.t. h ( x i ) = h ( x j ) if d ( x i , x j ) ≤ δ . √ Low dimensions: grid cells give d -approximation. Not quite a solution.

Locality Preserving Hashing Find nearby points in high dimensional space. Points could be images! Hash function h ( · ) s.t. h ( x i ) = h ( x j ) if d ( x i , x j ) ≤ δ . √ Low dimensions: grid cells give d -approximation. Not quite a solution. Why?

Locality Preserving Hashing Find nearby points in high dimensional space. Points could be images! Hash function h ( · ) s.t. h ( x i ) = h ( x j ) if d ( x i , x j ) ≤ δ . √ Low dimensions: grid cells give d -approximation. Not quite a solution. Why? Close to grid boundary.