# Squared norms: difference or notational convenience

• onako
In summary, when given a matrix A with dimensions n x m and asked to find a rank d approximation L that minimizes the Frobenius norm of the difference ||A-L||, the solution can be obtained through singular value decomposition of A, where only the d dominant singular values are kept and the rest are set to zero. However, there may be cases where we need to minimize an adapted norm, such as ||A-K||^2 or \left(\frac{||A-K||}{||A||}\right)^{1/2}, which may result in different solutions compared to the non-squared Frobenius norm. This is due to potentially confusing monotone transformations applied to the norm function f(K). Ultimately, all
onako
Given certain matrix $$A\in\mathbb{R}^{n\times m},$$the rank d approximation L with the same number of rows/column as A, minimizing the Frobenius norm of the difference $$||A-L||$$ is matrix obtained by singular value decomposition of A, with only d dominant singular values (the rest is simply set to zero).

However, I often encounter the minimization of the adapted norm, such as various kinds of normalization on the norm, ie.
$$i) ||A-K||^2$$
$$ii) \left(\frac{||A-K||}{||A||}\right)^{1/2}$$
and I'm not sure if the solution L from the above non-squared Frobenius norm coincides with the normalized Frobenius norm solution from i) and ii).
Isn't it the case that K should be L, but appropriately scaled for i) and/or ii)?

Last edited:
Essentially you're given a function f(K) and asked to minimize it. You're then asked to minimize f(K)^2 and f(K)/constant. All of these functions have the same minimum because the operations you are applying to f are all monotone

Thanks; I had similar reasoning. However, I'm surprised that in the literature one might find some confusing monotone transformations.

## 1. What is the difference between squared norms and regular norms?

Squared norms and regular norms both measure the magnitude of a vector or the size of a matrix. However, squared norms are calculated by squaring each element in the vector or matrix and then taking the sum, while regular norms are calculated by taking the square root of the sum of squared elements. This means that squared norms will always result in a positive value, while regular norms can be negative.

## 2. Are squared norms just a notational convenience?

Yes, squared norms are often used as a notational convenience because they are easier to work with mathematically. For example, squaring a vector or matrix simplifies certain calculations and can also make the results more interpretable.

## 3. How do squared norms relate to the Pythagorean Theorem?

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This concept can also be applied to squared norms, where the squared norm of a vector is equivalent to the sum of the squares of its individual components.

## 4. Can squared norms be negative?

No, squared norms will always result in a positive value because each element is squared before being summed. This means that even if the original vector or matrix contains negative values, the squared norm will always be positive.

## 5. What is the significance of squared norms in statistics?

Squared norms are used in statistics to calculate the variance of a data set. The squared norm of a data set is equal to the sum of squared deviations from the mean, which is a crucial component in calculating the variance. Squared norms are also used in other statistical calculations, such as in regression analysis and in determining the distance between data points.

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