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Viewed 2k times. Farhana Liza Farhana Liza 23 6 6 bronze badges. Add a comment. Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. The downside is that I could also simply use the words without fully understanding the underlying idea, while my expert listener filled in the gaps. As an exercise, I want to first present the SVD without jargon, as if explaining it to an interested year-old—think eighth-grade level mathematical maturity.
I will then formalize this intuitive explanation to work towards the standard formulation. So here we go. Imagine we have a square. We can manipulate this square in certain ways. For example, we can pull or push on an edge to stretch or compress the square Figure 2A and 2B. The only constraint is that our transformation must be linear. Intuitively, a linear transformation is one in which a straight line before the transformation results in a straight line after the transformation.
To visualize what this means, imagine a grid of evenly spaced vertical and horizontal lines on our square. A linear transformation will be one such that after the transformation, this diagonal line is still straight Figure 3B. To imagine the nonlinear transformation in Figure 3C, imagine bending a piece of engineering paper by pushing two sides together so that it bows in the middle.
Consider any linear transformation M M M , which we will apply to our square. If we are allowed to rotate our square before applying M M M , then we can find a rotation such that, by first applying the rotation and then applying M M M , we transform our square into a rectangle.
In other words, if we rotate the square before applying M M M , then M M M just stretches, compresses, or flips our square. We can avoid our square being sheared.
Imagine we sheared our square by pushing horizontally on its top left corner Figure 4A. The result is a sheared square, kind of like an old barn tipping over. But if we rotated our square before pushing it sideways, the shear would result in only stretching and compressing the square, albeit in a new orientation Figure 4B.
This is the geometric essence of SVD. Any linear transformation can be thought of as simply stretching or compressing or flipping a square, provided we are allowed to rotate it first. The transformed square or rectangle may have a new orientation after the transformation. Why is this a useful thing to do? For example, if one of the singular values is 0 0 0 , this means that our transformation flattens our square. Figure 5. What does Figure 5 represent geometerically?
The larger of the two singular values is the length of major axis of the ellipse. And since we transformed a perfect circle, every possible radii the edge of the circle has been stretched to the edge of the new ellipse. You can see that in the previous example. Consider this transformation:. This will have the effect of transforming the unit sphere into an ellipsoid :.
Its singular values are 3, 2, and 1. You can see how they again form the semi-axes of the resulting figure. The resulting figure now lives in a 2-dimensional space. Set it to A singular value and its singular vectors give the direction of maximum action among all directions orthogonal to the singular vectors of any larger singular value.
This has important applications.
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