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Matrix values ​​increasing after SVD, singular value decomposition

I am trying to learn SVD for image processing ... for example, compression.

My approach: get the image as a BufferedImage using ImageIO ... get the RGB values ​​and use them to get the equivalent grayscale value (which is between 0-255) and save it in an array double[][]. And use this array in SVD to compress it.

I get my USV matrices correctly ... hopefully. I get from U from AATranspose (AAT) and V from ATA.

Let me give you an example.

A is my original matrix.

A = 7.0     3.0     2.0
    9.0     7.0     5.0
    9.0     8.0     7.0
    5.0     3.0     6.0

U = -0.34598    -0.65267    -0.59969    -0.30771
    -0.57482    -0.27634     0.26045     0.72484
    -0.64327     0.21214     0.44200    -0.58808
    -0.36889     0.67280    -0.61415     0.18463

S = 21.57942    0.00000    0.00000
     0.00000    3.35324    0.00000
     0.00000    0.00000    2.02097
     0.00000    0.00000    0.00000

VT = -0.70573    -0.52432    -0.47649
     -0.53158    -0.05275     0.84536
     -0.46838     0.84989    -0.24149

So, now I need to make an extension of the external product, leaving a few terms for compression. Let's call the shortened members k.

k = 1 , ,

B = 6.43235    4.03003    1.70732
    9.24653    6.55266    5.12711
    9.41838    7.24083    7.21571
    4.41866    4.05485    5.70027

, B (, , SVD) , .

A . , 0-255. > 255 .

?

EDIT: k - , . , k = 1, , , :

A = u1 * S11 * vt1 + u2 * S22 + vt2

u1 u2 - 1 2 U, vt1 vt2 - 1 2 V.

+4
1

, Kaggle. , ... , , , A' S. , k=1, :

21.57942    0.00000    0.00000
0.00000     0.00000    0.00000
0.00000     0.00000    0.00000
0.00000     0.00000    0.00000

A', , A , ​​ A' = US'Vt. A':

5.268951 3.914582 3.557461
8.753958 6.503777 5.910449
9.796523 7.278353 6.614362
5.617932 4.173858 3.793084

, A, , , U, Vt , S.

: .

+1

Source: https://habr.com/ru/post/1611319/

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