Entire Frame Image Display Employing Monotonic Convergent Non-negative Matrix Factorization Yogesh Kumar Soniwal Supervisors: Prof. K. S. Venkatesh, Department of Electrical Engineering and Prof. Amit Mitra, Department of Mathematics and Statistics
Department of Electrical Engineering Indian Institute of Technology Kanpur May 6, 2014
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
2 / 49
Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
2 / 49
Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
2 / 49
Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
2 / 49
Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
2 / 49
Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
2 / 49
Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
2 / 49
Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
2 / 49
Contents
Overview of the thesis Methodology used (SNMSR) Approach-I and its limitations Approach-II (Monotonic non-negative matrix factorization (NNMF)) Comparison of Approach-I and Approach-II
Randomized Monotonic NNMF Mid-point Monotonic NNMF Monotonic converging matrix series representation Conclusion Future work
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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The Why: Diode Lifespan Diminution in OLED Displays
Horizontal lines (Rows)
Vertical lines (Columns)
Figure: Schematic diagram of an OLED display
Intensity of (m, n)th pixel: η Driving (m, n)th pixel: Ground mth row for t time and supply l current at nth column, such that l×t=η Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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The What: Entire Frame Image Display
Figure: Illustration: Entire Frame Image Display
Representation of a non-negative matrix I in terms of W1 × H1 + . . . + WK × HK is termed as Separable Non-negative Matrix Series Representation (SNMSR) of I. Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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The How: NNMF and SNMSR
k ← 0
≤ ⃪ ⃪ ⃪
⃪
⃪
Figure: Sub-frame computation - “Approach-I” Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Notations
k th partial sum Pk =
k X
Iir
i=1
k th residue Jk = I − Pk Note: J0 = I is the initial (zeroth ) residue.
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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NNMF NNMF: Approximation of a non-negative matrix IM ×N as W × H WM ×p and Hp×N both are of rank p ≤ min{M, N }. Problem Formulation: Minimize mean square error according to the following convex optimization problem ([Lee et. al, 1999]) [W, H] = argmin ||I − W × H||2F subject to W ≥ 0, H ≥ 0 The Frobenius norm ||AP ×Q ||F of matrix A is defined as v u P Q uX X ||A||F = t |aij |2 i=1 j=1
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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NNMF NNMF: Approximation of a non-negative matrix IM ×N as W × H WM ×p and Hp×N both are of rank p ≤ min{M, N }. Problem Formulation: Minimize mean square error according to the following convex optimization problem ([Lee et. al, 1999]) [W, H] = argmin ||I − W × H||2F subject to W ≥ 0, H ≥ 0 The Frobenius norm ||AP ×Q ||F of matrix A is defined as v u P Q uX X ||A||F = t |aij |2 i=1 j=1
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Alternating Least Square-NNMF (ALS-NNMF)
ALS-NNMF ([Berry et. al, 2007]) Initialize W as random matrix of size M × p for i = 1 to iter do Solve H from W T W H = W T I Set all negative elements in H to 0 Solve W from HH T W T = HI T Set all negative elements in W to 0 end for
Yogesh Kumar Soniwal (IIT Kanpur)
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Overshoots Removal: Reduction Process (Step-2)
Example:
i11 i12 i13 I = i21 i22 i23 i31 i32 i33 w1 a11 a12 a13 I1 = w2 × h1 h2 h3 = a21 a22 a23 w3 a31 a32 a33 Problem: A lot of superfluous reduction process takes place. Solution: Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Overshoots Removal: Reduction Process (Step-2)
Example:
i11 i12 i13 I = i21 i22 i23 i31 i32 i33 w1 a11 a12 a13 I1 = w2 × h1 h2 h3 = a21 a22 a23 w3 a31 a32 a33 Problem: A lot of superfluous reduction process takes place. Solution: Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Overshoots Removal: Reduction Process (Step-2)
Example:
i11 i12 i13 I = i21 i22 i23 i31 i32 i33 w1 a11 a12 a13 I1 = w2 × h1 h2 h3 = a21 a22 a23 0 a32 a33 a31 w3
Problem: A lot of superfluous reduction process takes place. Solution: Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Overshoots Removal: Reduction Process (Step-2)
Example:
i11 i12 i13 I = i21 i22 i23 i31 i32 i33 w1 a 11 a12 a13 0 a22 a23 a21 I1 = w2 × h1 h2 h3 = 0 a33 a31 a32 w3
Problem: A lot of superfluous reduction process takes place. Solution: Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Overshoots Removal: Reduction Process (Step-2)
Example:
i11 i12 i13 I = i21 i22 i23 i31 i32 i33 a13 w1 a a12 11 0 0 a21 a a23 I1 = w2 × h1 h2 h3 = 22 0 a33 a32 a31 w3
Problem: A lot of superfluous reduction process takes place. Solution: Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Overshoots Removal: Reduction Process (Step-2)
Example:
i11 i12 i13 I = i21 i22 i23 i31 i32 i33 a13 w1 a a12 11 0 0 a21 a a23 I1 = w2 × h1 h2 h3 = 22 0 a33 a32 a31 w3
Problem: A lot of superfluous reduction process takes place. Solution: Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF
Problem Formulation: Given a non-negative matrix IM ×N , find two matrices WM ×1 and H1×N , that approximate I to W × H as closely as possible, subject to the following constraints on W and H: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , M }, ∀ s = {1, . . . , N } and
wr hs ≤ irs ∀ r = {1, . . . , M } & ∀ s = {1, . . . , N }. Third constraints are monotonic constraints. Initialization: Choose h1 ∈ [0, i11 ] randomly.
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF
Problem Formulation: Given a non-negative matrix IM ×N , find two matrices WM ×1 and H1×N , that approximate I to W × H as closely as possible, subject to the following constraints on W and H: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , M }, ∀ s = {1, . . . , N } and
wr hs ≤ irs ∀ r = {1, . . . , M } & ∀ s = {1, . . . , N }. Third constraints are monotonic constraints. Initialization: Choose h1 ∈ [0, i11 ] randomly.
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
10 / 49
Monotonic NNMF
Problem Formulation: Given a non-negative matrix IM ×N , find two matrices WM ×1 and H1×N , that approximate I to W × H as closely as possible, subject to the following constraints on W and H: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , M }, ∀ s = {1, . . . , N } and
wr hs ≤ irs ∀ r = {1, . . . , M } & ∀ s = {1, . . . , N }. Third constraints are monotonic constraints. Initialization: Choose h1 ∈ [0, i11 ] randomly.
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
i13 i23 i33 .. .
··· ··· ··· .. .
i1N i2N i3N .. .
iM 1 iM 2 iM 3
···
iM N
i11 i21 i31 .. .
i12 i22 i32 .. .
Local constraints corresponding to the block matrix are: w1 ≥ 0, h1 ≥ 0 and w1 h1 ≤ i11 . From above equation we get, w1 = Yogesh Kumar Soniwal (IIT Kanpur)
i11 . h1
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
i13 i23 i33 .. .
··· ··· ··· .. .
i1N i2N i3N .. .
iM 1 iM 2 iM 3
···
iM N
i11 i21 i31 .. .
i12 i22 i32 .. .
Local constraints corresponding to the block matrix are: w1 ≥ 0, h1 ≥ 0 and w1 h1 ≤ i11 . From above equation we get, w1 = Yogesh Kumar Soniwal (IIT Kanpur)
i11 . h1
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
i13 i23 i33 .. .
··· ··· ··· .. .
i1N i2N i3N .. .
iM 1 iM 2 iM 3
···
iM N
i11 i21 i31 .. .
i12 i22 i32 .. .
Local constraints corresponding to the block matrix are: w1 ≥ 0, h1 ≥ 0, h2 ≥ 0, w1 h1 ≤ i11 and w1 h2 ≤ i12 . Here, the new variable is h2 . From monotonic constraints of h2 corresponding to this block matrix we get, h2 = Yogesh Kumar Soniwal (IIT Kanpur)
i12 . w1
Monotonic NNMF & SNMSR
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Monotonic NNMF Algorithm
i13 i23 i33 .. .
··· ··· ··· .. .
i1N i2N i3N .. .
iM 1 iM 2 iM 3
···
iM N
i11 i21 i31 .. .
i12 i22 i32 .. .
Local constraints corresponding to the block matrix are: w1 ≥ 0, h1 ≥ 0, h2 ≥ 0, w1 h1 ≤ i11 and w1 h2 ≤ i12 . Here, the new variable is h2 . From monotonic constraints of h2 corresponding to this block matrix we get, h2 = Yogesh Kumar Soniwal (IIT Kanpur)
i12 . w1
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
i13 i23 i33 .. .
··· ··· ··· .. .
i1N i2N i3N .. .
iM 1 iM 2 iM 3
···
iM N
i11 i21 i31 .. .
i12 i22 i32 .. .
Local constraints corresponding to the block matrix are: w1 ≥ 0, w2 ≥ 0, h1 ≥ 0, h2 ≥ 0, w1 h1 ≤ i11 , w1 h2 ≤ i12 , w2 h1 ≤ i21 and w2 h2 ≤ i22 . Here, the new variable is w2 . From monotonic constraints of w2 corresponding to this block matrix we get, i21 i22 , . w2 = min h1 h2 Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
i13 i23 i33 .. .
··· ··· ··· .. .
i1N i2N i3N .. .
iM 1 iM 2 iM 3
···
iM N
i11 i21 i31 .. .
i12 i22 i32 .. .
Local constraints corresponding to the block matrix are: w1 ≥ 0, w2 ≥ 0, h1 ≥ 0, h2 ≥ 0, w1 h1 ≤ i11 , w1 h2 ≤ i12 , w2 h1 ≤ i21 and w2 h2 ≤ i22 . Here, the new variable is w2 . From monotonic constraints of w2 corresponding to this block matrix we get, i21 i22 , . w2 = min h1 h2 Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
i13 i23 i33 .. .
··· ··· ··· .. .
i1N i2N i3N .. .
iM 1 iM 2 iM 3
···
iM N
i11 i21 i31 .. .
i12 i22 i32 .. .
Local constraints corresponding to the block matrix are: w1 ≥ 0, w2 ≥ 0, h1 ≥ 0, h2 ≥ 0, h3 ≥ 0, w1 h1 ≤ i11 , w1 h2 ≤ i12 , w1 h3 ≤ i13 , w2 h1 ≤ i21 , w2 h2 ≤ i22 and w2 h3 ≤ i23 . Here, the new variable is h3 . From monotonic constraints of h3 corresponding to this block matrix we get, i13 i23 h3 = min , . w1 w2 Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
i13 i23 i33 .. .
··· ··· ··· .. .
i1N i2N i3N .. .
iM 1 iM 2 iM 3
···
iM N
i11 i21 i31 .. .
i12 i22 i32 .. .
Local constraints corresponding to the block matrix are: w1 ≥ 0, w2 ≥ 0, h1 ≥ 0, h2 ≥ 0, h3 ≥ 0, w1 h1 ≤ i11 , w1 h2 ≤ i12 , w1 h3 ≤ i13 , w2 h1 ≤ i21 , w2 h2 ≤ i22 and w2 h3 ≤ i23 . Here, the new variable is h3 . From monotonic constraints of h3 corresponding to this block matrix we get, i13 i23 h3 = min , . w1 w2 Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
i13 i23 i33 .. .
··· ··· ··· .. .
i1N i2N i3N .. .
iM 1 iM 2 iM 3
···
iM N
i11 i21 i31 .. .
i12 i22 i32 .. .
Local constraints corresponding to the block matrix are: w1 ≥ 0, w2 ≥ 0, h1 ≥ 0, h2 ≥ 0, h3 ≥ 0, w1 h1 ≤ i11 , w1 h2 ≤ i12 , w1 h3 ≤ i13 , w2 h1 ≤ i21 , w2 h2 ≤ i22 and w2 h3 ≤ i23 . Here, the new variable is h3 . From monotonic constraints of h3 corresponding to this block matrix we get, i13 i23 h3 = min , . w1 w2 Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm A general step: 1. Vertical Traversal i11 ··· i1k i1,k+1 .. . .. . .. .. . . ik−1,1 · · · ik−1,k ik−1,k+1 ik1 ··· ikk ik,k+1 ik+1,1 · · · ik+1,k ik+1,k+1 .. .. .. .. . . . . iM 1 ··· iM k iM,k+1
··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Local constraints corresponding to the block matrix are: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , k}, ∀ s = {1, . . . , k} and
wr hs ≤ irs ∀ r = {1, . . . , k} & ∀ s = {1, . . . , k}. Yogesh Kumar Soniwal (IIT Kanpur)
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Monotonic NNMF Algorithm A general step: 1. Vertical Traversal i11 ··· i1k i1,k+1 .. . .. . .. .. . . ik−1,1 · · · ik−1,k ik−1,k+1 ik1 ··· ikk ik,k+1 ik+1,1 · · · ik+1,k ik+1,k+1 .. .. .. .. . . . . iM 1 ··· iM k iM,k+1
··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Local constraints corresponding to the block matrix are: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , k}, ∀ s = {1, . . . , k} and
wr hs ≤ irs ∀ r = {1, . . . , k} & ∀ s = {1, . . . , k}. Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
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Monotonic NNMF Algorithm A general step: 1. Vertical Traversal i11 ··· i1k i1,k+1 .. . .. . .. .. . . ik−1,1 · · · ik−1,k ik−1,k+1 ik1 ··· ikk ik,k+1 ik+1,1 · · · ik+1,k ik+1,k+1 .. .. .. .. . . . . iM 1 ··· iM k iM,k+1
··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Local constraints corresponding to the block matrix are: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , k}, ∀ s = {1, . . . , k} and
wr hs ≤ irs ∀ r = {1, . . . , k} & ∀ s = {1, . . . , k}. Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm A general step: 1. Vertical Traversal i11 ··· i1k i1,k+1 .. . .. . .. .. . . ik−1,1 · · · ik−1,k ik−1,k+1 ik1 ··· ikk ik,k+1 ik+1,1 · · · ik+1,k ik+1,k+1 .. .. .. .. . . . . iM 1 ··· iM k iM,k+1
··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Local constraints corresponding to the block matrix are: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , k}, ∀ s = {1, . . . , k} and
wr hs ≤ irs ∀ r = {1, . . . , k} & ∀ s = {1, . . . , k}. Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm A general step: 1. Vertical Traversal i11 ··· i1k i1,k+1 .. . .. . .. .. . . ik−1,1 · · · ik−1,k ik−1,k+1 ik1 ··· ikk ik,k+1 ik+1,1 · · · ik+1,k ik+1,k+1 .. .. .. .. . . . . iM 1 ··· iM k iM,k+1
··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Here, the new variable is wk . From monotonic constraints of wk corresponding to this block matrix we get, ik1 ikk wk = min ,..., . h1 hk Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm A general step: 1. Vertical Traversal i11 ··· i1k i1,k+1 .. . .. . .. .. . . ik−1,1 · · · ik−1,k ik−1,k+1 ik1 ··· ikk ik,k+1 ik+1,1 · · · ik+1,k ik+1,k+1 .. .. .. .. . . . . iM 1 ··· iM k iM,k+1
··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Here, the new variable is wk . From monotonic constraints of wk corresponding to this block matrix we get, ik1 ikk wk = min ,..., . h1 hk Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm A general step: 2. Horizontal Traversal i11 ··· i1,k−1 i1k i1,k+1 .. . . .. . .. .. .. . . ik−1,1 · · · ik−1,k−1 ik−1,k ik−1,k+1 ik1 ··· ik,k−1 ikk ik,k+1 .. . . .. . .. .. .. . . iM 1 ··· iM,k−1 iM k iM,k+1
··· .. . ··· ··· .. . ···
i1N .. . ik−1,N ikN .. . iM N
Local constraints corresponding to the block matrix are: wr ≥ 0
∀ r = {1, . . . , k − 1},
hs ≥ 0
∀ s = {1, . . . , k} and
wr hs ≤ irs ∀ r = {1, . . . , k − 1} & ∀ s = {1, . . . , k}. Yogesh Kumar Soniwal (IIT Kanpur)
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Monotonic NNMF Algorithm A general step: 2. Horizontal Traversal i11 ··· i1,k−1 i1k i1,k+1 .. . . .. . .. .. .. . . ik−1,1 · · · ik−1,k−1 ik−1,k ik−1,k+1 ik1 ··· ik,k−1 ikk ik,k+1 .. . . .. . .. .. .. . . iM 1 ··· iM,k−1 iM k iM,k+1
··· .. . ··· ··· .. . ···
i1N .. . ik−1,N ikN .. . iM N
Local constraints corresponding to the block matrix are: wr ≥ 0
∀ r = {1, . . . , k − 1},
hs ≥ 0
∀ s = {1, . . . , k} and
wr hs ≤ irs ∀ r = {1, . . . , k − 1} & ∀ s = {1, . . . , k}. Yogesh Kumar Soniwal (IIT Kanpur)
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Monotonic NNMF Algorithm A general step: 2. Horizontal Traversal i11 ··· i1,k−1 i1k i1,k+1 .. . . .. . .. .. .. . . ik−1,1 · · · ik−1,k−1 ik−1,k ik−1,k+1 ik1 ··· ik,k−1 ikk ik,k+1 .. . . .. . .. .. .. . . iM 1 ··· iM,k−1 iM k iM,k+1
··· .. . ··· ··· .. . ···
i1N .. . ik−1,N ikN .. . iM N
Local constraints corresponding to the block matrix are: wr ≥ 0
∀ r = {1, . . . , k − 1},
hs ≥ 0
∀ s = {1, . . . , k} and
wr hs ≤ irs ∀ r = {1, . . . , k − 1} & ∀ s = {1, . . . , k}. Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm A general step: 2. Horizontal Traversal i11 ··· i1,k−1 i1k i1,k+1 .. . . .. . .. .. .. . . ik−1,1 · · · ik−1,k−1 ik−1,k ik−1,k+1 ik1 ··· ik,k−1 ikk ik,k+1 .. . . .. . .. .. .. . . iM 1 ··· iM,k−1 iM k iM,k+1
··· .. . ··· ··· .. . ···
i1N .. . ik−1,N ikN .. . iM N
Local constraints corresponding to the block matrix are: wr ≥ 0
∀ r = {1, . . . , k − 1},
hs ≥ 0
∀ s = {1, . . . , k} and
wr hs ≤ irs ∀ r = {1, . . . , k − 1} & ∀ s = {1, . . . , k}. Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
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Monotonic NNMF Algorithm A general step: 2. Horizontal Traversal i11 ··· i1,k−1 i1k i1,k+1 .. . . .. . .. .. .. . . ik−1,1 · · · ik−1,k−1 ik−1,k ik−1,k+1 ik1 ··· ik,k−1 ikk ik,k+1 .. . . .. . .. .. .. . . iM 1 ··· iM,k−1 iM k iM,k+1
··· .. . ··· ··· .. . ···
i1N .. . ik−1,N ikN .. . iM N
Here, the new variable is hk . From monotonic constraints of hk corresponding to this block matrix we get, ik−1,k i1k hk = min ,..., . w1 wk−1
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Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm A general step: 2. Horizontal Traversal i11 ··· i1,k−1 i1k i1,k+1 .. . . .. . .. .. .. . . ik−1,1 · · · ik−1,k−1 ik−1,k ik−1,k+1 ik1 ··· ik,k−1 ikk ik,k+1 .. . . .. . .. .. .. . . iM 1 ··· iM,k−1 iM k iM,k+1
··· .. . ··· ··· .. . ···
i1N .. . ik−1,N ikN .. . iM N
Here, the new variable is hk . From monotonic constraints of hk corresponding to this block matrix we get, ik−1,k i1k hk = min ,..., . w1 wk−1
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Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
14 / 49
Monotonic NNMF Algorithm
Example: i11 i21 i31 i41
i12 i22 i32 i42
i13 i23 i33 i43
Yogesh Kumar Soniwal (IIT Kanpur)
i14 i24 i34 i44
i11 i21 i31 i41
i12 i22 i11 i12 i13 i14 i32 i21 i22 i23 i24 i42
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm The last step in a square matrix i11 ··· .. .. . . iN −1,1 · · · iN 1 ···
i1N .. .
iN −1,N iN N
Local constraints corresponding to the block matrix are: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , N }, ∀ s = {1, . . . , N } and
wr hs ≤ irs ∀ r = {1, . . . , N } & ∀ s = {1, . . . , N }.
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm The last step in a square matrix i11 ··· .. .. . . iN −1,1 · · · iN 1 ···
i1N .. .
iN −1,N iN N
Local constraints corresponding to the block matrix are: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , N }, ∀ s = {1, . . . , N } and
wr hs ≤ irs ∀ r = {1, . . . , N } & ∀ s = {1, . . . , N }.
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
15 / 49
Monotonic NNMF Algorithm The last step in a square matrix i11 ··· .. .. . . iN −1,1 · · · iN 1 ···
i1N .. .
iN −1,N iN N
Local constraints corresponding to the block matrix are: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , N }, ∀ s = {1, . . . , N } and
wr hs ≤ irs ∀ r = {1, . . . , N } & ∀ s = {1, . . . , N }.
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
15 / 49
Monotonic NNMF Algorithm The last step in a square matrix i11 ··· .. .. . . iN −1,1 · · · iN 1 ···
i1N .. .
iN −1,N iN N
Local constraints corresponding to the block matrix are: wr ≥ 0 hs ≥ 0
∀ r = {1, . . . , N }, ∀ s = {1, . . . , N } and
wr hs ≤ irs ∀ r = {1, . . . , N } & ∀ s = {1, . . . , N }.
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
Horizontal only traversal step in a fat matrix: i11 · · · i1,k−1 i1k i1,k+1 · · · .. .. .. .. .. .. . . . . . . iM 1 · · · iM,k−1 iM k iM,k+1 · · ·
i1N .. . iM N
Similar to a horizontal traversal, we get iM k i1k ,..., . hk = min w1 wM
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
Horizontal only traversal step in a fat matrix: i11 · · · i1,k−1 i1k i1,k+1 · · · .. .. .. .. .. .. . . . . . . iM 1 · · · iM,k−1 iM k iM,k+1 · · ·
i1N .. . iM N
Similar to a horizontal traversal, we get iM k i1k hk = min ,..., . w1 wM
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
Horizontal only traversal step in a fat matrix: i11 · · · i1,k−1 i1k i1,k+1 · · · .. .. .. .. .. .. . . . . . . iM 1 · · · iM,k−1 iM k iM,k+1 · · ·
i1N .. . iM N
Similar to a horizontal traversal, we get iM k i1k hk = min ,..., . w1 wM
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
16 / 49
Monotonic NNMF Algorithm
Horizontal only traversal step in a fat matrix: i11 · · · i1,k−1 i1k i1,k+1 · · · .. .. .. .. .. .. . . . . . . iM 1 · · · iM,k−1 iM k iM,k+1 · · ·
i1N .. . iM N
Similar to a horizontal traversal, we get iM k i1k hk = min ,..., . w1 wM
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm
Horizontal only traversal step in a fat matrix: i11 · · · i1,k−1 i1k i1,k+1 · · · .. .. .. .. .. .. . . . . . . iM 1 · · · iM,k−1 iM k iM,k+1 · · ·
i1N .. . iM N
Similar to a horizontal traversal, we get i1k iM k hk = min ,..., . w1 wM
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm Vertical only traversal step in a i11 .. . ik−1,1 ik1 ik+1,1 .. . iM 1
fat matrix: ··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Similar to a vertical traversal, we get ik1 ikN wk = min ,..., . h1 hN
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Monotonic NNMF Algorithm Vertical only traversal step in a i11 .. . ik−1,1 ik1 ik+1,1 .. . iM 1
fat matrix: ··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Similar to a vertical traversal, we get ik1 ikN wk = min ,..., . h1 hN
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
17 / 49
Monotonic NNMF Algorithm Vertical only traversal step in a i11 .. . ik−1,1 ik1 ik+1,1 .. . iM 1
fat matrix: ··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Similar to a vertical traversal, we get ik1 ikN wk = min ,..., . h1 hN
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
17 / 49
Monotonic NNMF Algorithm Vertical only traversal step in a i11 .. . ik−1,1 ik1 ik+1,1 .. . iM 1
fat matrix: ··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Similar to a vertical traversal, we get ik1 ikN wk = min ,..., . h1 hN
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
17 / 49
Monotonic NNMF Algorithm Vertical only traversal step in a i11 .. . ik−1,1 ik1 ik+1,1 .. . iM 1
fat matrix: ··· .. . ··· ··· ··· .. . ···
i1N .. . ik−1,N ikN ik+1,N .. . iM N
Similar to a vertical traversal, we get ik1 ikN wk = min ,..., . h1 hN
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
17 / 49
Computational Complexity of ALS-NNMF
ALS-NNMF: Computational Complexity Calculate H from W T W H = W T I WT I → O MN WTW → O M
(W T W )−1 → O 1 Overall (in calculating H): O M N + O M + O Overall (in calculating W ): O N M + O N + O
1 ∼ O MN 1 ∼ O MN Computational complexity of ALS-NNMF: O 2 × iter × M N
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Computational Complexity of Monotonic NNMF
Square Matrix # of division operations in computing wk : k, ∀k = {1, . . . , N } # of division operations in computing hk : k − 1, ∀k = {1, . . . , N } Hence total # of operations: N X k=1
Yogesh Kumar Soniwal (IIT Kanpur)
k+
N X (k − 1) = N 2 k=1
Monotonic NNMF & SNMSR
May 6, 2014
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Computational Complexity of Monotonic NNMF Tall Matrix # of division operations in computing hk : k − 1, ∀k = {1, . . . , N } # of division operations in computing wk : k, ∀k = {1, . . . , N } # of division operations in computing wk : N , ∀k = {N + 1, . . . , M } Hence total # of operations: N X k=1
(k − 1) +
N X
k + N (M − N ) = M N
k=1
Fat Matrix From symmetry, total operations: M N (by interchanging M and N )
Computational Complexity of Monotonic NNMF Hence computational complexity of Monotonic NNMF is O M N Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Time Elapsed Comparison of ALS and Monotonic NNMF Time elapsed comparison of ALS−NNMF and Monotonic NNMF (Excluding reduction process) 0.09 ALS−NNMF Monotonic−NNMF 0.08
0.07
Time(sec.)
0.06
0.05
0.04
0.03
0.02
0.01
100
200
300
400
500 600 Matrix Size (n)
700
800
900
1000
Figure: Time elapsed comparison for ALS-NNMF and Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Approach-II of Sub-frame Computation
k ← 0
⃪
⃪
Figure: Sub-frame computation - “Approach-II (Monotonic NNMF)”
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Time Elapsed in One Sub-frame Computation Time elapsed comparison of Approach−I and Approach−II for SNMSR computation 0.02 1
0.018
0.016
0
10
0.014
0.012
−1
10
0.01 −2
0.008
10
0.006 −3
10
0.004
Approach−II (Monotonic NNMF) − Time(sec.)
Approach−I (ALS−NNMF followed by reduction) − Time(sec.)
10
0.002 −4
10
0
100
200
300
400
500 600 Matrix Size (n)
700
800
900
0 1000
Figure: Time elapsed in one sub-frame computation for Approach-I and Approach-II
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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RMSE Comparison of Approach-I And Approach-II
RMSE RMSE in first sub-frame (I1 = W × H) computation is defined as s ||I − W H||2F RMSE = MN
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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RMSE Comparison of Approach-I and Approach-II RMSE in first sub−frame computation for Approach−I and Approach−II 150
Approach−I (ALS−NNMF followed by reduction) Approach−II (Monotonic NNMF)
RMSE
100
50
0
0
100
200
300
400
500 Matrix Size (n)
600
700
800
900
1000
Figure: RMSE in first sub-frame computation Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
130 31 12 102 46 230 I= 19 61 240 61 106 125 11.4 2.72 1.05 11.4 130 31 12 8.95 102 24.3 9.4 I1 = W1 × H1 = 1.67 19 4.5 1.8 5.35 61 14.5 5.6 0 0 0 0 21.7 220.6 ⇒ J1 = I − I1 = 0 56.5 238.2 0 91.5 119.4
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
130 31 12 102 46 230 I= 19 61 240 61 106 125 11.4 2.72 1.05 11.4 130 31 12 8.95 102 24.3 9.4 I1 = W1 × H1 = 1.67 19 4.5 1.8 5.35 61 14.5 5.6 0 0 0 0 21.7 220.6 ⇒ J1 = I − I1 = 0 56.5 238.2 0 91.5 119.4
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
130 31 12 102 46 230 I= 19 61 240 61 106 125 11.4 2.72 1.05 11.4 130 31 12 8.95 102 24.3 9.4 I1 = W1 × H1 = 1.67 19 4.5 1.8 5.35 61 14.5 5.6 0 0 0 0 21.7 220.6 ⇒ J1 = I − I1 = 0 56.5 238.2 0 91.5 119.4
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
0 0 0 0 21.7 220.6 J1 = 0 56.5 238.2 0 91.5 119.4 0 0 0 24.62 0 I2 = W2 × H2 = 26.6 0 13.32 0
0.88 8.96 0 0 21.7 220.6 23.4 238.2 11.7 119.4 0 0 0 0 0 0 ⇒ J 2 = I − I1 − I2 = 0 33.1 0 0 79.8 0
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
0 0 0 0 21.7 220.6 J1 = 0 56.5 238.2 0 91.5 119.4 0 0 0 24.62 0 I2 = W2 × H2 = 26.6 0 13.32 0
0.88 8.96 0 0 21.7 220.6 23.4 238.2 11.7 119.4 0 0 0 0 0 0 ⇒ J 2 = I − I1 − I2 = 0 33.1 0 0 79.8 0
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
0 0 0 0 21.7 220.6 J1 = 0 56.5 238.2 0 91.5 119.4 0 0 0 24.62 0 I2 = W2 × H2 = 26.6 0 13.32 0
0.88 8.96 0 0 21.7 220.6 23.4 238.2 11.7 119.4 0 0 0 0 0 0 ⇒ J 2 = I − I1 − I2 = 0 33.1 0 0 79.8 0
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
0 0 0 0 0 0 J2 = 0 33.1 0 0 79.8 0 0 0 0 0 0 I3 = W3 × H3 = 1.59 0 3.84 0
20.77 0 0 33.1 79.8 0 0 0 0 ⇒ J3 = I − I1 − I2 − I3 = 0 0 0 0
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
0 0 0 0 0 0 0 0 0 May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
0 0 0 0 0 0 J2 = 0 33.1 0 0 79.8 0 0 0 0 0 0 I3 = W3 × H3 = 1.59 0 3.84 0
20.77 0 0 33.1 79.8 0 0 0 0 ⇒ J3 = I − I1 − I2 − I3 = 0 0 0 0
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
0 0 0 0 0 0 0 0 0 May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
0 0 0 0 0 0 J2 = 0 33.1 0 0 79.8 0 0 0 0 0 0 I3 = W3 × H3 = 1.59 0 3.84 0
20.77 0 0 33.1 79.8 0 0 0 0 ⇒ J3 = I − I1 − I2 − I3 = 0 0 0 0
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
0 0 0 0 0 0 0 0 0 May 6, 2014
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SNMSR Employing Monotonic NNMF Example:
130 31 12 102 46 230 I= 19 61 240 = 61 106 125 0 0.88 0 0 0 24.62 0 21.7 26.6 0 23.4 13.32 0 11.7
Yogesh Kumar Soniwal (IIT Kanpur)
11.4 2.72 1.05 11.4 130 31 12 8.95 102 24.3 9.4 + 1.67 19 4.5 1.8 5.35 61 14.5 5.6
8.96 0 220.6 + 238.2 119.4
0 0 0 0 0 1.59 0 3.84 0
Monotonic NNMF & SNMSR
20.77 0 0 0 0 0 33.1 0 79.8 0
May 6, 2014
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SNMSR Employing Monotonic NNMF
Figure: Images used for simulation
These images are taken from the [USC-SIPI] image database. Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Simulation Results of SNMSR Employing Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Simulation Results of SNMSR Employing Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s irs ir,s−1 ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Randomized Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
ir−2,s−2 ir−1,s−2 ir,s−2 ir+1,s−2 ir+2,s−2 .. .
.. .
ir−2,s−1 ir−2,s ir−1,s−1 ir−1,s ir,s−1 irs ir+1,s−1 ir+1,s ir+2,s−1 ir+2,s .. .. . .
ir,s
ir,s−2
.. .
.. .
ir−2,s+1 ir−2,s+2 ir−1,s+1 ir−1,s+2 ir,s+1 ir,s+2 ir+1,s+1 ir+1,s+2 ir+2,s+1 ir+2,s+2 .. .. . .
ir,s+2
··· ···
ir+2,s ir+2,s−2 ir+2,s+2 ir−2,s ir−2,s−2 ir−2,s+2
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Simulation Results of SNMSR Employing Randomized Monotonic NNMF
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Simulation Results of SNMSR Employing Randomized Monotonic NNMF
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Energy Covered vs Sub-frame Count Energy of an Image E(I) =
M X N X
i2mn
m=1 n=1
Energy of k th partial sum (Pk ) E(Pk ) =
M X N X
p2kmn
m=1 n=1
Percentage energy covered in k th partial sum E(Pk ) =
Yogesh Kumar Soniwal (IIT Kanpur)
E(Pk ) × 100 E(I)
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Energy Covered vs Sub-frame Count For SNMSR Employing Monotonic NNMF Energy Covered With Each Partial Sum Using Monotonic NNMF
90 elaine lena baboon vegetables texture1 texture2 nature text
80
Percentage Energy Covered
70
60
50
40
30
20
10
50
100
150
200
250 300 Partial Sum Index
350
400
450
500
Figure: Energy covered in each partial sum for SNMSR using Monotonic NNMF Yogesh Kumar Soniwal (IIT Kanpur)
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Energy Covered vs Sub-frame Count For SNMSR Employing Randomized Monotonic NNMF Energy Covered With Each Partial Sum Using Randomized Monotonic NNMF
90
80
Percentage Energy Covered
70
elaine lena baboon vegetables texture1 texture2 nature text
60
50
40
30
20
10
50
100
150 Partial Sum Index
200
250
Figure: Energy covered in each partial sum for SNMSR using Randomized Monotonic NNMF Yogesh Kumar Soniwal (IIT Kanpur)
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Sub-frames Required for convergence vs Matrix Size Number of sub−frames required for convergence of SNMSR vs Matrix Size
900
800
Monotonic NNMF Randomized Monotonic NNMF Linear approximation of first 100 values of Randomized Monotonic NNMF curve
Sub−frames Required
700
600
500
400
300
200
100
100
200
300
400
500 Matrix Size (n)
600
700
800
900
Figure: Number of sub-frames required for convergence of SNMSR vs matrix size Yogesh Kumar Soniwal (IIT Kanpur)
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Reason for Improvement in Sub-frame Count After Randomization Number of elements exactly factorized in each sub−frame for lena image of size 512 × 512
1400
SNMSR using Monotonic NNMF SNMSR using Randomized Monotonic NNMF 2N−1 = 1023
Number of elements exactly factorized
1200
1000
800
600
400
200
0
0
100
200
300 Sub−frame count
400
500
600
Figure: Number of elements exactly factorized in each sub-frame for lena image of size 512 × 512 for SNMSR using Monotonic and Randomized Monotonic NNMF Yogesh Kumar Soniwal (IIT Kanpur)
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Driving Time and DTR Driving Time Driving time of a display is defined as the time required to drive the entire display ([Eisenbrand et. al, 2009]). [Driving Time]Traditional ∝
M X
max imn
m=1
[Driving Time]Proposed ∝
K X
1≤n≤N
max akmn
1≤m≤M k=1 1≤n≤N
Driving Time Ratio We define the ratio of driving time of traditional approach and our approach as Driving Time Ratio (DTR). Yogesh Kumar Soniwal (IIT Kanpur)
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Driving Time Example:
130 31 12 102 46 230 = I= 19 61 240 61 106 125
11.4 2.72 11.4 130 31 8.95 102 24.3 1.67 19 4.5 5.35 61 14.5
0 0.88 8.96 0 0 0 0 24.62 0 21.7 220.6 + 26.6 0 23.4 238.2 13.32 0 11.7 119.4
0 0 0 0 0 1.59 0 3.84 0
1.05 12 9.4 + 1.8 5.6 20.77 0 0 33.1 79.8
0 0 0 0 0
[Driving Time]Traditional = 130 + 230 + 240 + 125 = 725 units. [Driving Time]Proposed = 130 + 238.2 + 79.8 = 448 units. Yogesh Kumar Soniwal (IIT Kanpur)
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Simulation Results of DTR Driving Time Ratio vs Matrix Size
3.2
3
Driving Time Ratio
2.8
Monotonic NNMF Randomized Monotonic NNMF
2.6
2.4
2.2
2
1.8
1.6 100
200
300
400
500 Matrix Size
600
700
800
900
Figure: DTR vs matrix size
Improvement of more than 50% in driving time for proposed scheme Yogesh Kumar Soniwal (IIT Kanpur)
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A Variation of Monotonic NNMF: Mid-point Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
.. .
i N −2, N −1 i N −2, N i N −2, N −2 2 2 2 2 2 2 i N −1, N −2 i N −1, N −1 i N −1, N 2 2 2 2 2 2 i N , N −2 i N , N −1 iN ,N 2 2 2 2 2 2 i N +1, N −2 i N +1, N −1 i N +1, N 2 2 2 2 2 2 i N +2, N −1 i N +2, N i N +2, N −2 2 2 2 2 2 2 .. .. .. . . .
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
.. . i N −2, N +1 2 2 i N −1, N +1 2 2 i N , N +1 2 2 i N +1, N +1 2 2 i N +2, N +1 2 2 .. .
··· ··· ··· ··· ··· .. .
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A Variation of Monotonic NNMF: Mid-point Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
.. .
i N −2, N −1 i N −2, N i N −2, N −2 2 2 2 2 2 2 i N −1, N −2 i N −1, N −1 i N −1, N 2 2 2 2 2 2 i N , N −2 i N , N −1 iN ,N 2 2 2 2 2 2 i N +1, N −2 i N +1, N −1 i N +1, N 2 2 2 2 2 2 i N +2, N −1 i N +2, N i N +2, N −2 2 2 2 2 2 2 .. .. .. . . .
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
.. . i N −2, N +1 2 2 i N −1, N +1 2 2 i N , N +1 2 2 i N +1, N +1 2 2 i N +2, N +1 2 2 .. .
··· ··· ··· ··· ··· .. .
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A Variation of Monotonic NNMF: Mid-point Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
.. .
i N −2, N −1 i N −2, N i N −2, N −2 2 2 2 2 2 2 i N −1, N −2 i N −1, N −1 i N −1, N 2 2 2 2 2 2 i N , N −2 i N , N −1 iN ,N 2 2 2 2 2 2 i N +1, N −2 i N +1, N −1 i N +1, N 2 2 2 2 2 2 i N +2, N −1 i N +2, N i N +2, N −2 2 2 2 2 2 2 .. .. .. . . .
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
.. . i N −2, N +1 2 2 i N −1, N +1 2 2 i N , N +1 2 2 i N +1, N +1 2 2 i N +2, N +1 2 2 .. .
··· ··· ··· ··· ··· .. .
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A Variation of Monotonic NNMF: Mid-point Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
.. .
i N −2, N −1 i N −2, N i N −2, N −2 2 2 2 2 2 2 i N −1, N −2 i N −1, N −1 i N −1, N 2 2 2 2 2 2 i N , N −2 i N , N −1 iN ,N 2 2 2 2 2 2 i N +1, N −2 i N +1, N −1 i N +1, N 2 2 2 2 2 2 i N +2, N −1 i N +2, N i N +2, N −2 2 2 2 2 2 2 .. .. .. . . .
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
.. . i N −2, N +1 2 2 i N −1, N +1 2 2 i N , N +1 2 2 i N +1, N +1 2 2 i N +2, N +1 2 2 .. .
··· ··· ··· ··· ··· .. .
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A Variation of Monotonic NNMF: Mid-point Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
.. .
i N −2, N −1 i N −2, N i N −2, N −2 2 2 2 2 2 2 i N −1, N −2 i N −1, N −1 i N −1, N 2 2 2 2 2 2 i N , N −2 i N , N −1 iN ,N 2 2 2 2 2 2 i N +1, N −2 i N +1, N −1 i N +1, N 2 2 2 2 2 2 i N +2, N −1 i N +2, N i N +2, N −2 2 2 2 2 2 2 .. .. .. . . .
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
.. . i N −2, N +1 2 2 i N −1, N +1 2 2 i N , N +1 2 2 i N +1, N +1 2 2 i N +2, N +1 2 2 .. .
··· ··· ··· ··· ··· .. .
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A Variation of Monotonic NNMF: Mid-point Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
.. .
i N −2, N −1 i N −2, N i N −2, N −2 2 2 2 2 2 2 i N −1, N −2 i N −1, N −1 i N −1, N 2 2 2 2 2 2 i N , N −2 i N , N −1 iN ,N 2 2 2 2 2 2 i N +1, N −2 i N +1, N −1 i N +1, N 2 2 2 2 2 2 i N +2, N −2 i N +2, N −1 i N +2, N 2 2 2 2 2 2 .. .. .. . . .
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
.. . i N −2, N +1 2 2 i N −1, N +1 2 2 i N , N +1 2 2 i N +1, N +1 2 2 i N +2, N +1 2 2 .. .
··· ··· ··· ··· ··· .. .
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A Variation of Monotonic NNMF: Mid-point Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
.. .
i N −2, N −1 i N −2, N i N −2, N −2 2 2 2 2 2 2 i N −1, N −2 i N −1, N −1 i N −1, N 2 2 2 2 2 2 i N , N −2 i N , N −1 iN ,N 2 2 2 2 2 2 i N +1, N −2 i N +1, N −1 i N +1, N 2 2 2 2 2 2 i N +2, N −2 i N +2, N −1 i N +2, N 2 2 2 2 2 2 .. .. .. . . .
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
.. . i N −2, N +1 2 2 i N −1, N +1 2 2 i N , N +1 2 2 i N +1, N +1 2 2 i N +2, N +1 2 2 .. .
··· ··· ··· ··· ··· .. .
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A Variation of Monotonic NNMF: Mid-point Monotonic NNMF
··· ··· ··· ··· ···
.. .
.. .
.. .
i N −2, N −2 i N −2, N −1 i N −2, N 2 2 2 2 2 2 i N −1, N −2 i N −1, N −1 i N −1, N 2 2 2 2 2 2 i N , N −2 i N , N −1 iN ,N 2 2 2 2 2 2 i N +1, N −2 i N +1, N −1 i N +1, N 2 2 2 2 2 2 i N +2, N −2 i N +2, N −1 i N +2, N 2 2 2 2 2 2 .. .. .. . . .
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
.. . i N −2, N +1 2 2 i N −1, N +1 2 2 i N , N +1 2 2 i N +1, N +1 2 2 i N +2, N +1 2 2 .. .
··· ··· ··· ··· ··· .. .
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Simulation Results of SNMSR Employing Mid-point Monotonic NNMF
Figure: Simulation results of SNMSR employing Mid-point Monotonic NNMF
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Conclusion
Simulation results of SNMSR using Monotonic NNMF, Randomized Monotonic NNMF and Mid-point Monotonic NNMF indicate that SNMSR using Randomized Monotonic NNMF is likely to be converged in the least number of sub-frames among all such possibilities.
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Monotonic Converging Matrix Series Representation (MSR) Sign Matrix Approach We define SI as the sign matrix corresponding to I, whose (m, n)th elements smn is given by, 1 if imn ≥ 0 smn = −1 if imn < 0 We define another matrix AI corresponding to I, whose (m, n)th element amn is given by, imn if imn ≥ 0 amn = |imn | if imn < 0 Note that, I = AI . ∗ SI , where (.∗) is element wise multiplication.
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Monotonic Converging Matrix Series Representation (MSR)
MSR Using Sign Matrix AI =
K X
Ik
k=1
Since AI is a monotonic and converging series, this property also holds for AI . ∗ SI . Hence monotonic converging MSR for matrix I is given by, I=
K X
Ik . ∗ SI
k=1
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Monotonic Converging Matrix Series Representation (MSR)
Separate Series Approach We define two matrices AI and BI , corresponding to I, whose (m, n)th elements amn and bmn respectively are given by, imn if imn ≥ 0 amn = 0 if imn < 0 and
bmn =
Yogesh Kumar Soniwal (IIT Kanpur)
0 if imn ≥ 0 |imn | if imn < 0
Monotonic NNMF & SNMSR
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Monotonic Converging Matrix Series Representation (MSR) MSR Using Separate Series Thus, we express I as I = AI − BI . Since AI and BI are both non-negative matrices, we obtain their SNMSRs using Monotonic NNMF:
AI =
K X
0
Ik , BI =
k=1
K X
0
Ik
k=1
∵ AI and BI are both monotonic converging series, monotonic converging MSR of I is given by, I = AI − BI =
K X
0
Ik −
k=1 Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
K X
0
Ik
k=1 May 6, 2014
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Gains
Computationally less complex Monotonic NNMF Entire frame image display approach to drive the entire display using sub-frames Proposed approach is not restricted to certain specific class of images Significant improvement in driving time Significant improvement in lifespan of diodes (The exact quantum of improvement is not presented here)
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
46 / 49
Gains
Computationally less complex Monotonic NNMF Entire frame image display approach to drive the entire display using sub-frames Proposed approach is not restricted to certain specific class of images Significant improvement in driving time Significant improvement in lifespan of diodes (The exact quantum of improvement is not presented here)
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
46 / 49
Gains
Computationally less complex Monotonic NNMF Entire frame image display approach to drive the entire display using sub-frames Proposed approach is not restricted to certain specific class of images Significant improvement in driving time Significant improvement in lifespan of diodes (The exact quantum of improvement is not presented here)
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
46 / 49
Gains
Computationally less complex Monotonic NNMF Entire frame image display approach to drive the entire display using sub-frames Proposed approach is not restricted to certain specific class of images Significant improvement in driving time Significant improvement in lifespan of diodes (The exact quantum of improvement is not presented here)
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
46 / 49
Gains
Computationally less complex Monotonic NNMF Entire frame image display approach to drive the entire display using sub-frames Proposed approach is not restricted to certain specific class of images Significant improvement in driving time Significant improvement in lifespan of diodes (The exact quantum of improvement is not presented here)
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
46 / 49
Future Work
Parallelization of Monotonic NNMF Higher rank Monotonic NNMF Use image correlation in Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
47 / 49
Future Work
Parallelization of Monotonic NNMF Higher rank Monotonic NNMF Use image correlation in Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
47 / 49
Future Work
Parallelization of Monotonic NNMF Higher rank Monotonic NNMF Use image correlation in Monotonic NNMF
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
47 / 49
References
M. W. Berry, M. Browne, A. N. Langville, V. P. Pauca and R. J. Plemmons (2007) Algorithms and applications for approximate nonnegative matrix factorization Computational Statistics & Data Analysis 52(1), 155 – 173. D. D. Lee and H. S. Seung (1999) Learning the parts of objects by non-negative matrix factorization Nature 401(6755), 788–791. F. Eisenbrand, A. Karrenbauer and C. Xu (2009) Algorithms for longer OLED lifetime Journal of Experimental Algorithmics (JEA) 14(3).
The USC-SIPI Image Database [Online]. Available: http://sipi.usc.edu/database
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
48 / 49
Thank you :)
Yogesh Kumar Soniwal (IIT Kanpur)
Monotonic NNMF & SNMSR
May 6, 2014
49 / 49
