For a truncation of SVD to approximate a data matrix or to project onto modes, where , one uses the cumulative sum of the normalized energy defined as as an indicator of how much energy modes include.

Question

For a truncation of SVD to approximate a data matrix  or to project onto  modes, where , one uses the cumulative sum of the normalized energy defined as  as an indicator of how much energy  modes include.

BlackTom AI · Accepted Answer

The question discusses truncation of SVD and how to measure how much information (energy) is captured by the top k modes using an energy-based indicator.
First, recall the standard idea: the energy present in each singular value s_i is proportional to s_i^2. The total energy in the data (under SVD) is the sum of s_i^2 over all singular values considered. To quantify how much energy is captured by the first k modes, we typically compute the fraction E_k = (sum_{i=1}^k s_i^2) / (sum_{i=1}^r s_i^2), where r is the rank or number of singular values kept. This E_k is the cumulative energy explained by the first k modes, and it serves as an indicator of how much information those modes preserve.

Option critique:
- The statement as presented (incomplete in the prompt) suggests using the cumulative sum of the normalized energy as an indicator. If interpreted as summing s_i^2 normalized by the total energy to obtain a fraction, this aligns with the standard energy-explained criterion. In that reading, the approach is correct, because the normalized energy fractions add up to 1 and their cumulative sum up to k gives the proportion of total energy captured by the first k modes.
- A potential source of the stated falsehood is if the phrase "normalized energy" is misdefined or if one confuses energy with another metric (for example, using the sum of s_i without squaring, or normalizing by a quantity other than the total energy). In those misinterpretations, the indicator would be incorrect, since energy in SVD is tied to squared singular values, and meaningful energy fractions come from s_i^2, not from s_i directly.
- Another subtle point is that when projecting onto modes or choosing a truncation level, practitioners typically rely on the energy explained by the first k squared singular values divided by the total squared singular values. If the statement attempted to define the indicator using a different normalization or a different metric (e.g., not using squared values or not normalizing by total energy), that would render the claim false.

In summary, the core concept—using the cumulative sum of normalized energy (where energy per mode is proportional to s_i^2, and normalization is by the total s_i^2) to indicate how much energy the top k modes capture—is the standard and correct approach in many treatments of SVD truncation. However, ambiguity or an incorrect interpretation of what counts as "normalized energy" (especially if not using s_i^2 or not dividing by the total energy) could lead to a false statement. The given answer marks this as false, which would be consistent if the statement intended a definition that deviates from the conventional energy-based criterion (i.e., not using squared singular values or not normalizing by total energy).

AMATH 482 A Checkpoint 2 Quiz

For a truncation of SVD to approximate a data matrix or to project onto modes, where , one uses the cumulative sum of the normalized energy defined as as an indicator of how much energy modes include.

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