Tensor by Tensor

MODULE 01 · LESSON 07

Singular value decomposition

SVD ranks matrix directions. A moving average smooths changing measurements.

PLAIN-LANGUAGE INTRODUCTION

What is this?

SVD ranks matrix directions. A moving average smooths changing measurements.

One simple example

First use A=[[3,0],[0,1]], whose SVD strengths are 3 and 1. A later matrix has strengths 5 and 1. The leading-strength average becomes 3.5.

What goes in?

A matrix for SVD or a sequence of measurements for averaging.

What comes out?

Ranked direction strengths or one smoother measurement.

Why does it matter?

SVD exposes strong structure. Moving averages reduce short-term noise.

What is it not?

Neither method explains why the data changed.

WORK THROUGH THE IDEA

See the idea in more detail

  1. Singular Value Decomposition (SVD) writes a matrix as A = UΣVᵀ. The diagonal of Σ stores strengths.
  2. For A=[[3,0],[0,1]], the two singular values are 3 and 1. The first direction is stronger.
  3. A later matrix [[5,0],[0,1]] has singular values 5 and 1. Its leading strength changed from 3 to 5.
  4. Smooth that same measurement with decay 0.75: 0.75×3 + 0.25×5 = 3.5. These matrices use chosen values.
  5. Common mistake: a small singular value is not automatically useless. Its value depends on the task.
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