library(HSAUR2)Loading required package: toolsdata(USairpollution) Activity 3.1 - SVD
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Problem 1
Reconsider the US air pollution data set:
A)
Perform singular value decomposition of this data matrix. Then create the matrix \(D\). Describe what this matrix looks like.
library(tidyverse)── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr 1.1.4 ✔ readr 2.1.5
✔ forcats 1.0.0 ✔ stringr 1.5.1
✔ ggplot2 4.0.0 ✔ tibble 3.3.0
✔ lubridate 1.9.4 ✔ tidyr 1.3.1
✔ purrr 1.1.0
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errorshead(USairpollution) SO2 temp manu popul wind precip predays
Albany 46 47.6 44 116 8.8 33.36 135
Albuquerque 11 56.8 46 244 8.9 7.77 58
Atlanta 24 61.5 368 497 9.1 48.34 115
Baltimore 47 55.0 625 905 9.6 41.31 111
Buffalo 11 47.1 391 463 12.4 36.11 166
Charleston 31 55.2 35 71 6.5 40.75 148components <- svd(USairpollution)
U <- components$u
U %>% round(2) [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] -0.02 0.10 0.21 -0.25 0.01 -0.07 -0.04
[2,] -0.03 0.15 0.00 0.04 -0.19 -0.40 0.03
[3,] -0.09 0.08 0.11 0.11 -0.06 0.13 -0.02
[4,] -0.16 0.13 -0.02 -0.19 -0.11 0.10 0.07
[5,] -0.09 0.06 0.20 0.11 0.44 -0.11 0.15
[6,] -0.01 0.09 0.25 -0.05 0.09 -0.04 -0.37
[7,] -0.67 -0.44 -0.20 0.01 -0.04 0.05 -0.08
[8,] -0.09 -0.01 0.19 0.16 0.12 -0.02 -0.28
[9,] -0.18 -0.24 0.32 0.03 0.08 -0.12 -0.01
[10,] -0.08 0.19 0.05 -0.14 0.14 -0.01 -0.10
[11,] -0.15 0.07 -0.04 0.29 -0.13 -0.03 0.16
[12,] -0.10 0.01 0.07 0.13 -0.01 -0.35 -0.03
[13,] -0.03 0.10 0.13 0.04 0.02 -0.07 0.28
[14,] -0.26 0.17 -0.17 -0.17 0.21 -0.06 0.00
[15,] -0.06 -0.15 0.35 0.02 -0.12 0.08 0.03
[16,] -0.20 0.26 -0.19 0.10 -0.01 0.14 0.09
[17,] -0.11 0.24 -0.05 -0.19 0.07 0.05 0.07
[18,] -0.07 0.28 -0.02 0.01 -0.07 0.21 -0.08
[19,] -0.09 0.07 0.07 0.16 0.00 0.02 0.14
[20,] -0.02 0.07 0.17 0.22 -0.12 0.15 -0.03
[21,] -0.09 0.20 0.02 -0.14 0.01 0.09 -0.09
[22,] -0.10 0.18 0.00 0.13 -0.02 0.18 0.03
[23,] -0.06 0.12 0.16 0.29 -0.06 0.18 -0.17
[24,] -0.13 0.06 0.06 0.07 0.24 -0.10 0.24
[25,] -0.15 -0.02 0.13 0.03 0.26 -0.17 0.07
[26,] -0.08 0.13 0.09 0.09 0.01 0.10 -0.14
[27,] -0.06 0.13 0.11 0.25 -0.07 0.22 -0.10
[28,] -0.04 0.18 0.10 -0.09 -0.10 0.09 0.15
[29,] -0.06 0.12 0.07 0.05 0.01 -0.08 0.24
[30,] -0.37 -0.05 -0.14 -0.13 -0.07 0.10 -0.02
[31,] -0.08 0.22 -0.15 0.06 -0.37 -0.44 -0.32
[32,] -0.09 0.12 0.13 -0.38 0.00 -0.04 -0.06
[33,] -0.05 -0.09 0.32 -0.39 -0.38 0.10 0.21
[34,] -0.05 0.09 0.14 0.04 -0.06 0.06 -0.15
[35,] -0.03 0.05 0.14 -0.02 -0.10 -0.31 -0.02
[36,] -0.12 0.13 -0.08 0.10 -0.12 -0.19 0.02
[37,] -0.09 0.11 0.16 -0.08 0.29 -0.06 -0.15
[38,] -0.14 -0.16 0.21 0.08 -0.21 -0.03 0.02
[39,] -0.12 0.19 -0.03 -0.10 -0.03 0.04 0.01
[40,] -0.04 0.12 0.06 0.15 -0.09 -0.08 0.43
[41,] -0.02 0.05 0.22 -0.03 -0.12 0.04 0.03head(U) [,1] [,2] [,3] [,4] [,5] [,6]
[1,] -0.01841623 0.10181630 0.2091659528 -0.25110469 0.006626339 -0.06864647
[2,] -0.03130764 0.14812091 -0.0004043841 0.04012803 -0.190598264 -0.40490988
[3,] -0.08887282 0.08371317 0.1077274000 0.11286840 -0.060174208 0.13096949
[4,] -0.15620922 0.13142656 -0.0218222317 -0.18626470 -0.107291147 0.10220939
[5,] -0.08773289 0.05962107 0.1957870856 0.10730132 0.441622268 -0.10577782
[6,] -0.01291040 0.08879269 0.2527249241 -0.05135988 0.090737537 -0.03524838
[,7]
[1,] -0.03732041
[2,] 0.03112816
[3,] -0.02364724
[4,] 0.07385071
[5,] 0.14913872
[6,] -0.37039600D <- components$d
D %>% round(2)[1] 7051.95 931.12 540.46 92.71 85.24 52.95 10.14head(D)[1] 7051.94936 931.12109 540.46297 92.70909 85.23724 52.94650V <- components$v
V %>% round(2) [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] -0.03 0.00 0.21 -0.82 -0.54 0.03 0.01
[2,] -0.03 0.20 0.30 0.50 -0.67 -0.39 -0.11
[3,] -0.65 -0.71 0.26 0.09 0.00 -0.01 0.00
[4,] -0.75 0.57 -0.33 -0.07 0.01 0.01 0.00
[5,] -0.01 0.03 0.05 0.04 -0.04 -0.11 0.99
[6,] -0.02 0.13 0.24 0.24 -0.22 0.91 0.06
[7,] -0.07 0.34 0.79 -0.11 0.47 -0.12 -0.04#Specifically looking at the matrix D after the decomposition
D[1] 7051.94936 931.12109 540.46297 92.70909 85.23724 52.94650 10.14090#The component matrix of D has 7 columns of numerical values with only 1 row
#by implementing the component that basically takes the diagonal values of
#7x7 matrix (V). B)
Verify that \(X=UDV^T\) by plotting all the entries of \(X\) versus all the entries of \(UDV^T\) with the 0/1 line.
#Slides with the
USairpollution SO2 temp manu popul wind precip predays
Albany 46 47.6 44 116 8.8 33.36 135
Albuquerque 11 56.8 46 244 8.9 7.77 58
Atlanta 24 61.5 368 497 9.1 48.34 115
Baltimore 47 55.0 625 905 9.6 41.31 111
Buffalo 11 47.1 391 463 12.4 36.11 166
Charleston 31 55.2 35 71 6.5 40.75 148
Chicago 110 50.6 3344 3369 10.4 34.44 122
Cincinnati 23 54.0 462 453 7.1 39.04 132
Cleveland 65 49.7 1007 751 10.9 34.99 155
Columbus 26 51.5 266 540 8.6 37.01 134
Dallas 9 66.2 641 844 10.9 35.94 78
Denver 17 51.9 454 515 9.0 12.95 86
Des Moines 17 49.0 104 201 11.2 30.85 103
Detroit 35 49.9 1064 1513 10.1 30.96 129
Hartford 56 49.1 412 158 9.0 43.37 127
Houston 10 68.9 721 1233 10.8 48.19 103
Indianapolis 28 52.3 361 746 9.7 38.74 121
Jacksonville 14 68.4 136 529 8.8 54.47 116
Kansas City 14 54.5 381 507 10.0 37.00 99
Little Rock 13 61.0 91 132 8.2 48.52 100
Louisville 30 55.6 291 593 8.3 43.11 123
Memphis 10 61.6 337 624 9.2 49.10 105
Miami 10 75.5 207 335 9.0 59.80 128
Milwaukee 16 45.7 569 717 11.8 29.07 123
Minneapolis 29 43.5 699 744 10.6 25.94 137
Nashville 18 59.4 275 448 7.9 46.00 119
New Orleans 9 68.3 204 361 8.4 56.77 113
Norfolk 31 59.3 96 308 10.6 44.68 116
Omaha 14 51.5 181 347 10.9 30.18 98
Philadelphia 69 54.6 1692 1950 9.6 39.93 115
Phoenix 10 70.3 213 582 6.0 7.05 36
Pittsburgh 61 50.4 347 520 9.4 36.22 147
Providence 94 50.0 343 179 10.6 42.75 125
Richmond 26 57.8 197 299 7.6 42.59 115
Salt Lake City 28 51.0 137 176 8.7 15.17 89
San Francisco 12 56.7 453 716 8.7 20.66 67
Seattle 29 51.1 379 531 9.4 38.79 164
St. Louis 56 55.9 775 622 9.5 35.89 105
Washington 29 57.3 434 757 9.3 38.89 111
Wichita 8 56.6 125 277 12.7 30.58 82
Wilmington 36 54.0 80 80 9.0 40.25 114X <- (U %*% diag(D) %*% t(V) %>% round(1))
head(X) [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 46 47.6 44 116 8.8 33.4 135
[2,] 11 56.8 46 244 8.9 7.8 58
[3,] 24 61.5 368 497 9.1 48.3 115
[4,] 47 55.0 625 905 9.6 41.3 111
[5,] 11 47.1 391 463 12.4 36.1 166
[6,] 31 55.2 35 71 6.5 40.7 148k <-2
#k=2
(X_tilde2 <- U[,1:k] %*% diag(D[1:k]) %*% t(V[,1:k]))%>%round(1) [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 3.5 23.8 17.2 151.5 3.6 15.2 42.1
[2,] 5.9 35.8 45.7 244.5 5.5 22.8 63.7
[3,] 17.3 37.8 352.3 516.8 6.2 24.6 73.1
[4,] 30.4 63.4 629.6 900.0 10.5 41.3 123.5
[5,] 17.1 32.9 363.0 498.0 5.5 21.5 64.8
[6,] 2.4 20.0 0.5 115.4 3.0 12.7 35.1
[7,] 131.5 80.9 3371.0 3334.4 17.1 56.6 208.0
[8,] 18.2 20.7 433.8 488.4 3.8 13.8 44.9
[9,] 34.5 -2.1 962.8 807.5 1.1 0.1 15.2
[10,] 16.2 56.9 260.0 548.2 9.0 36.6 104.9
[11,] 29.3 50.8 643.7 839.4 8.6 33.3 101.6
[12,] 19.2 26.4 443.0 528.5 4.6 17.4 54.9
[13,] 6.3 26.2 86.1 223.8 4.1 16.8 47.7
[14,] 50.9 96.7 1088.6 1481.9 16.2 63.2 190.8
[15,] 11.3 -15.3 362.9 221.1 -1.8 -9.2 -19.7
[16,] 38.8 98.3 746.5 1199.8 16.0 63.6 186.9
[17,] 22.1 73.6 368.8 736.7 11.7 47.4 136.3
[18,] 13.5 71.1 138.3 526.0 11.0 45.4 127.3
[19,] 17.7 35.8 369.9 520.6 6.0 23.4 70.1
[20,] 4.6 18.8 65.6 163.8 2.9 12.1 34.2
[21,] 17.7 60.9 289.1 595.9 9.6 39.2 112.5
[22,] 19.2 58.8 336.5 624.1 9.4 37.9 109.7
[23,] 11.0 36.3 183.1 364.5 5.8 23.4 67.3
[24,] 25.5 43.8 559.9 728.1 7.4 28.7 87.7
[25,] 28.5 32.6 680.9 766.8 5.9 21.8 70.5
[26,] 14.5 42.3 261.2 465.3 6.8 27.3 79.3
[27,] 11.4 38.9 186.7 382.2 6.2 25.0 71.8
[28,] 8.3 44.1 83.6 324.4 6.8 28.2 78.9
[29,] 10.6 37.2 171.2 359.4 5.9 23.9 68.6
[30,] 71.6 81.4 1712.1 1924.8 14.8 54.4 176.3
[31,] 15.9 62.2 233.5 555.9 9.8 39.9 113.6
[32,] 17.4 45.1 331.6 541.4 7.3 29.2 85.6
[33,] 10.4 -4.0 301.8 233.9 -0.2 -2.1 -1.1
[34,] 10.0 30.3 177.0 324.6 4.8 19.5 56.6
[35,] 6.4 17.3 118.4 200.1 2.8 11.2 32.7
[36,] 23.2 53.9 462.5 703.5 8.8 35.0 103.6
[37,] 18.2 44.1 358.0 558.0 7.2 28.6 84.3
[38,] 27.5 4.6 745.0 660.4 1.8 4.0 22.6
[39,] 23.7 65.8 438.8 751.3 10.6 42.5 123.9
[40,] 8.1 33.2 115.3 288.9 5.2 21.3 60.5
[41,] 3.4 13.3 49.9 118.9 2.1 8.5 24.3k <- 3
(X_tilde3<- U[,1:k] %*% diag(D[1:k]) %*% t(V[,1:k]))%>%round(1) [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] 27.4 58.3 46.0 114.5 9.8 42.3 131.8
[2,] 5.9 35.7 45.6 244.5 5.5 22.8 63.5
[3,] 29.6 55.5 367.1 497.7 9.4 38.5 119.3
[4,] 27.9 59.8 626.6 903.9 9.9 38.5 114.1
[5,] 39.5 65.1 390.0 463.4 11.3 46.8 148.8
[6,] 31.3 61.6 35.4 70.6 10.5 45.5 143.5
[7,] 109.1 48.7 3343.9 3369.1 11.3 31.2 123.9
[8,] 40.5 52.8 460.7 453.9 9.5 39.1 128.4
[9,] 70.9 50.2 1006.7 751.2 10.5 41.3 151.5
[10,] 22.1 65.4 267.1 539.1 10.5 43.3 127.0
[11,] 25.2 44.7 638.7 845.9 7.5 28.5 85.9
[12,] 27.3 38.0 452.8 515.9 6.7 26.6 85.4
[13,] 20.9 47.2 103.7 201.2 7.9 33.3 102.4
[14,] 31.6 68.9 1065.3 1511.8 11.2 41.3 118.5
[15,] 51.9 43.1 411.9 158.2 8.7 36.8 132.5
[16,] 16.9 66.9 720.2 1233.6 10.3 39.0 105.2
[17,] 16.9 66.1 362.5 744.8 10.3 41.5 116.8
[18,] 11.6 68.3 136.0 529.1 10.5 43.2 120.0
[19,] 25.8 47.6 379.7 508.0 8.1 32.6 100.7
[20,] 24.3 47.0 89.3 133.4 8.0 34.3 107.8
[21,] 20.2 64.5 292.1 592.1 10.3 42.0 121.7
[22,] 18.8 58.2 336.0 624.8 9.3 37.4 108.1
[23,] 28.9 62.1 204.7 336.8 10.4 43.6 134.4
[24,] 32.4 53.8 568.3 717.3 9.2 36.6 113.8
[25,] 43.2 53.8 698.7 744.0 9.7 38.4 125.7
[26,] 25.3 57.9 274.3 448.5 9.6 39.6 119.9
[27,] 24.2 57.2 202.1 362.5 9.5 39.4 119.6
[28,] 19.2 59.8 96.8 307.5 9.6 40.5 119.8
[29,] 18.4 48.4 180.6 347.3 7.9 32.7 97.7
[30,] 55.9 58.7 1693.1 1949.2 10.7 36.5 117.2
[31,] -1.5 37.2 212.5 582.8 5.2 20.2 48.4
[32,] 32.7 67.1 350.0 517.7 11.3 46.5 143.0
[33,] 47.3 49.0 346.3 176.9 9.4 39.6 137.0
[34,] 26.4 53.8 196.7 299.3 9.1 38.1 117.9
[35,] 21.9 39.7 137.1 176.0 6.8 28.8 91.0
[36,] 14.6 41.6 452.2 716.8 6.6 25.3 71.4
[37,] 36.1 69.8 379.6 530.3 11.8 48.8 151.3
[38,] 51.9 39.7 774.4 622.6 8.1 31.7 114.1
[39,] 20.4 61.0 434.8 756.4 9.7 38.7 111.4
[40,] 15.2 43.3 123.8 278.0 7.0 29.2 86.8
[41,] 28.6 49.5 80.3 79.9 8.6 37.0 118.6#plotting the entrie '0,1' abline for k =2
plot(X,X_tilde2);abline(0,1)
#plotting the '0,1' abline for k=3
plot(X,X_tilde3);abline(0,1)
C)
Consider low-dimensional approximations of the data matrix. What is the fewest number of dimensions required to yield a correlation between the entries of \(X\) and \(\tilde X\) of at least 0.9?
#Checking the correlation of the dim. approx. with k=2
cor(as.vector(X),as.vector(X_tilde2))[1] 0.9965504#Value is 0.99655
cor(as.vector(X),as.vector(X_tilde3))[1] 0.9997697#Value is 0.99977
#The fewest number of dimensions required is k = 2. D)
Find \(\Sigma\), the covariance matrix of this data set. Then perform eigen-decomposition of this matrix. Verify that
- The eigenvectors of \(\Sigma\) equal the columns of \(V\)
- The eigenvalues of \(\Sigma\) equal the diagonals of \(D^2/(n-1)\)
#
standardized_X <- scale(X, center = TRUE, scale = TRUE)
sigma <- cov(standardized_X)
sigma [,1] [,2] [,3] [,4] [,5] [,6]
[1,] 1.00000000 -0.43360020 0.64476873 0.49377958 0.09469045 0.05374692
[2,] -0.43360020 1.00000000 -0.19004216 -0.06267813 -0.34973963 0.38631564
[3,] 0.64476873 -0.19004216 1.00000000 0.95526935 0.23794683 -0.03284543
[4,] 0.49377958 -0.06267813 0.95526935 1.00000000 0.21264375 -0.02642512
[5,] 0.09469045 -0.34973963 0.23794683 0.21264375 1.00000000 -0.01262596
[6,] 0.05374692 0.38631564 -0.03284543 -0.02642512 -0.01262596 1.00000000
[7,] 0.36956363 -0.43024212 0.13182930 0.04208319 0.16410559 0.49598467
[,7]
[1,] 0.36956363
[2,] -0.43024212
[3,] 0.13182930
[4,] 0.04208319
[5,] 0.16410559
[6,] 0.49598467
[7,] 1.00000000#The eigen-decomposition of the matrix
components <- svd(sigma)
U_sigma <- components$u
D_sigma <- components$d
V_sigma <- components$v
U_sigma %>% round(2) [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] -0.49 -0.08 -0.01 0.40 -0.73 0.18 -0.15
[2,] 0.32 0.09 -0.68 -0.18 -0.16 0.61 0.02
[3,] -0.54 0.23 -0.27 -0.03 0.16 -0.04 0.75
[4,] -0.49 0.28 -0.34 -0.11 0.35 -0.09 -0.65
[5,] -0.25 -0.06 0.31 -0.86 -0.27 0.15 -0.02
[6,] 0.00 -0.63 -0.49 -0.18 -0.16 -0.55 0.01
[7,] -0.26 -0.68 0.11 0.11 0.44 0.50 -0.01D_sigma %>% round(2)[1] 2.73 1.51 1.39 0.89 0.35 0.10 0.03V_sigma %>% round(2) [,1] [,2] [,3] [,4] [,5] [,6] [,7]
[1,] -0.49 -0.08 -0.01 0.40 -0.73 0.18 -0.15
[2,] 0.32 0.09 -0.68 -0.18 -0.16 0.61 0.02
[3,] -0.54 0.23 -0.27 -0.03 0.16 -0.04 0.75
[4,] -0.49 0.28 -0.34 -0.11 0.35 -0.09 -0.65
[5,] -0.25 -0.06 0.31 -0.86 -0.27 0.15 -0.02
[6,] 0.00 -0.63 -0.49 -0.18 -0.16 -0.55 0.01
[7,] -0.26 -0.68 0.11 0.11 0.44 0.50 -0.01eig <-eigen(sigma)
eig_vectors <- eig$vectors
eig_values <- eig$values
#Confirm that eigenvectors of sigma equal to col(V)
all.equal(eig$vectors,V_sigma)[1] "Mean relative difference: 1.15259"#Not exactly equal has a mean relative diff. of 1.15
all.equal(eig$values, ((D_sigma)^2)/1)[1] "Mean relative difference: 0.925403"#Not exactly equal has a mean relative diff. of 0.93Problem 2
In this problem we explore how “high” a low-dimensional SVD approximation of an image has to be before you can recognize it.
.Rdata objects are essentially R workspace memory snapshots that, when loaded, load any type of R object that you want into your global environment. The command below, when executed, will load three objects into your memory: mysteryU4, mysteryD4, and mysteryV4. These are the first \(k\) vectors and singular values of an SVD I performed on a 700-pixels-tall \(\times\) 600-pixels-wide image of a well-known villain.
load('C:/Users/lr7273ow/OneDrive - Minnesota State/Documents/GitHub/DSCI_415/Activities/Data/mystery_person_k4.Rdata')
#Now we have mysteryU4, mysteryD4,mysteryV4A)
Write a function that takes SVD ingredients u, d and v and renders the \(700 \times 600\) image produced by this approximation using functions from the magick package. Use your function to determine whether a 4-dimensional approximation to this image is enough for you to tell who the mystery villain is. Recall that you will likely need to rescale your recomposed approximation so that all pixels are in [0,1].
#At a k = 4 approximation, it will look like this..
library(magick)Warning: package 'magick' was built under R version 4.5.2Linking to ImageMagick 6.9.13.29
Enabled features: cairo, freetype, fftw, ghostscript, heic, lcms, pango, raw, rsvg, webp
Disabled features: fontconfig, x11k<-4
Xtilde <- mysteryU4 %*% diag(mysteryD4) %*% t(mysteryV4)
#Scaling the Xtilde
Xtilde_scaled <- (Xtilde - min(Xtilde)) / (max(Xtilde) - min(Xtilde))
(Xtilde_scaled
%>% as.raster
%>% image_read)
#No, k = 4 is not enough for me to tell who the villain is based on a glance (tho I think I know who it is)B)
I’m giving you slightly higher-dimensional approximations (\(k=10\) and \(k=50\), respectively) in the objects below:
load('C:/Users/lr7273ow/OneDrive - Minnesota State/Documents/GitHub/DSCI_415/Activities/Data/mystery_person_k10.Rdata')
load('C:/Users/lr7273ow/OneDrive - Minnesota State/Documents/GitHub/DSCI_415/Activities/Data/mystery_person_k50.Rdata')Create both of the images produced by these approximations. At what point can you tell who the mystery villain is
#At a k = 10 approximation, it will look like this..
k<-10
X_tilde10 <- mysteryU10 %*% diag(mysteryD10) %*% t(mysteryV10)
Xtilde10_scaled <- (X_tilde10 - min(X_tilde10))/(max(X_tilde10) - min(X_tilde10))
(Xtilde10_scaled
%>% as.raster
%>% image_read
)
#Not quite but I believe the picture is todd iverson
#At a k = 50 approximation, it will look like this..
k<-50
X_tilde50 <- mysteryU50 %*% diag(mysteryD50) %*% t(mysteryV50)
Xtilde50_scaled <- (X_tilde50 - min(X_tilde50))/(max(X_tilde50) - min
(X_tilde50))
(Xtilde50_scaled
%>% as.raster
%>% image_read
)
#Yep, thats Todd Iverson :P.C)
How many numbers need to be stored in memory for each of the following:
- A full \(700\times 600\) image?
- A 4-dimensional approximation?
- A 10-dimensional approximation?
- A 50-dimensional approximation?
A full 700*600 image would require 420000 storage (total)
k = 4
U4 = 700 *4 = 2800, D4 = 4 = 16, V4 = 600*4 = 2400
2400+ 4 + 2800 = 5204
5204 would be the numbers required to store in memory
k=10
U4 = 700 *10 = 7000, D4 = 10*1 = 10, V4 = 600*10 = 6000
7000 + 100 + 6000 = 13010
k=50
U4 = 700 *50 = 35000, D4 = 50*1 = 10, V4 = 600*50 = 30000
35000 + 50 + 30000 = 650050