One of the compelling applications of Singular Value Decomposition (SVD) is in data compression, particularly for data that can be naturally represented as matrices, like images. The core idea relies on the property that SVD organizes the information within a matrix in order of importance, allowing us to discard less important information while retaining a good approximation of the original data.

Recall the SVD of an $m \times n$ matrix $A$ : $A = U \Sigma V^T$ Here, $\Sigma$ is an $m \times n$ diagonal matrix containing the singular values $\sigma_1, \sigma_2, \sigma_3, \dots$ ordered from largest to smallest ( $\sigma_1 \ge \sigma_2 \ge \dots \ge 0$ ). These singular values quantify the importance of the corresponding directions captured by the columns of $U$ and $V$ . Larger singular values correspond to dimensions that capture more of the variance or "energy" of the data in the matrix.

Data compression via SVD works by creating a low-rank approximation of the original matrix $A$ . Instead of using all the singular values and vectors, we keep only the first $k$ components, where $k$ is smaller than the rank of the original matrix. This process is often called truncated SVD.

We construct the approximation $A_k$ as follows: $A_k = U_k \Sigma_k V_k^T$ Where:

$U_k$ consists of the first $k$ columns of $U$ (an $m \times k$ matrix).
$\Sigma_k$ is a $k \times k$ diagonal matrix containing the first $k$ singular values ( $\sigma_1, \dots, \sigma_k$ ).
$V_k^T$ consists of the first $k$ rows of $V^T$ (which are the first $k$ columns of $V$ , transposed, resulting in a $k \times n$ matrix).

The Eckart-Young-Mirsky theorem states that this matrix $A_k$ is the best rank- $k$ approximation of $A$ in terms of the Frobenius norm (and the spectral norm). Essentially, by keeping the components associated with the largest singular values, we retain the most significant parts of the original matrix $A$ .

Why is this Compression?

Storing the original matrix $A$ requires storing $m \times n$ numbers. Storing the truncated SVD components $U_k$ , $\Sigma_k$ , and $V_k^T$ requires storing:

$m \times k$ values for $U_k$
$k$ values for the diagonal elements of $\Sigma_k$
$k \times n$ values for $V_k^T$

The total storage is $mk + k + kn = k(m + n + 1)$ values. If $k$ is significantly smaller than $m$ and $n$ , then $k(m + n + 1)$ can be much smaller than $m \times n$ , achieving compression.

Image Compression Example

A grayscale digital image can be represented as a matrix where each entry corresponds to the pixel intensity at a specific location. Let's say we have a $512 \times 512$ grayscale image. This matrix $A$ has $512 \times 512 = 262,144$ entries.

Apply SVD: Compute $A = U \Sigma V^T$ . $U$ and $V$ will be $512 \times 512$ , and $\Sigma$ will be $512 \times 512$ .
Truncate: Choose a value for $k$ . For example, let $k=50$ . Construct $A_{50} = U_{50} \Sigma_{50} V_{50}^T$ .
Store: Instead of storing the original 262,144 pixel values, we store $U_{50}$ ( $512 \times 50$ ), $\Sigma_{50}$ ( $50$ values), and $V_{50}^T$ ( $50 \times 512$ ). The total storage is $50 \times 512 + 50 + 50 \times 512 = 25600 + 50 + 25600 = 51,250$ values. This is roughly 20% of the original storage size.
Reconstruct: To view the compressed image, multiply the stored components: $A_{50} = U_{50} \Sigma_{50} V_{50}^T$ . The resulting matrix $A_{50}$ is an approximation of the original image $A$ .

The choice of $k$ determines the trade-off between compression ratio and image quality. A smaller $k$ yields higher compression but a potentially lower-quality image reconstruction. A larger $k$ results in better fidelity to the original image but less compression. Often, a relatively small $k$ can capture most of the visual information, as the singular values tend to decrease rapidly for typical images.

Singular values for an image matrix often show a steep initial drop followed by a long tail of smaller values. This suggests that keeping only the first few dozen or hundred singular values (corresponding to a rank- $k$ approximation) can preserve much of the image's structure while discarding components associated with smaller singular values, which might correspond to finer details or noise.

For color images, which typically have separate channels (e.g., Red, Green, Blue), SVD compression can be applied independently to the matrix representing each color channel.

Lossy Compression

It's important to note that SVD-based compression is lossy. The reconstructed matrix $A_k$ is not identical to the original matrix $A$ (unless $k$ equals the rank of $A$ ). Some information is permanently discarded during the truncation step. However, because SVD prioritizes the most significant components, the perceptual difference between $A$ and $A_k$ can be small, especially for moderate values of $k$ , making it effective for applications like image compression where perfect reconstruction is not always necessary.

While image compression is a classic illustration, the principle applies to any data representable as a matrix where a lower-rank approximation is acceptable and beneficial for storage or computational efficiency. This could include large feature matrices in machine learning, user-item interaction matrices in recommendation systems, or numerical simulation data.