How do you pick the truncated-SVD rank k for a users-by-items ratings matrix?

## Why there is no formula A users-by-items ratings matrix factored by a truncated SVD produces k latent factors per user and per item. The value of k controls three things at once: how much of the matrix's structure is retained, how many free numbers the representation carries and therefore how easily it fits noise, and how much it costs to store and serve. No single criterion addresses all three, which is why this is a judgment call and interviewers ask it as one. ## Step 1: let the spectrum bracket the range List the singular values in order. Real ratings matrices typically show a few large values followed by a long, slowly decaying tail. The point where the decay flattens is informative: layers beyond it each contribute a similar, small amount of energy, so they are hard to distinguish from noise. Treat this as a **bracket**, not an answer. Reading a precise k off a curve with no sharp elbow is the classic overreach, and many real spectra have no visible break at all. Eckart-Young converts any candidate k into an exact error figure: the Frobenius residual is `sqrt(sigma_{k+1}^2 + ... + sigma_r^2)`. That is useful for pricing the options, and it is free once you have the spectrum. ## Step 2: evaluate on data you did not factor Reconstruction error on the entries you factored is monotonically decreasing in k, so it always votes for the largest k and can never tell you to stop. The fix is a holdout: mask a random sample of known ratings, factor the remaining matrix at each candidate k, and score the predictions on the masked entries. That curve typically falls and then rises, because each extra layer adds `m + n + 1` free numbers and eventually starts reproducing noise in the observed ratings instead of structure. The turning point is the honest signal. Masking by random entries and masking whole users or items answer different questions, and a system that must serve new users should be evaluated the second way. ## Step 3: respect the operational budget A rank-k factorization stores `k(m + n + 1)` numbers. On a matrix with millions of users this dominates memory, the per-request latency of scoring a user against candidate items, and the wall-clock cost of refactoring on a schedule. A k that is twice as large for a fractional gain in held-out error is usually the wrong call. Break-even against storing the raw matrix arrives at `k(m + n + 1) = m n`, though for a large sparse ratings matrix the practical comparison is against the number of observed ratings, not against mn. ## Step 4: judge on the metric that matters Reconstruction error is a proxy. The system is graded on something else, and the mapping between the two is not monotone: a k that reconstructs held-out ratings slightly better may not change the ordering the product depends on. Sweep a small set of candidate k values against the downstream metric and prefer the smallest that is not meaningfully worse, since smaller is cheaper, faster to recompute and more stable across refits. ## The missing-data caveat A plain SVD needs a value in every cell, and a ratings matrix is mostly empty. Filling the gaps with zeros or a global mean makes the decomposition fit the fabricated values as though they were observations, which distorts both the spectrum and the choice of k: what you are looking at partly describes your imputation. This is why factorizations that model only the observed entries are preferred for this matrix. Naming that caveat unprompted is a strong signal in an interview, because it shows you know the difference between the mathematics of the SVD and the shape of real ratings data. ## Stability as a tiebreaker A final consideration: refit the factorization on a slightly different slice of data and see whether the chosen k still looks reasonable. If the held-out curve is flat over a wide band of k, any value in that band is defensible, and picking the low end is the safer engineering decision. ## Common mistakes Declaring a k from the spectrum alone; optimizing reconstruction error on the fitted entries; quoting a universal rule such as a fixed number of factors for every dataset; ignoring serving cost and refactoring cadence; and silently zero-filling missing ratings.