PCA with missing data

Ipsen & Hansen (2019). One planted direction, signal-to-noise S, α = N/D samples per dimension, each entry missing with probability m. The plot shows the squared overlap R² of the learned direction with the true one.

1. Three readings of "missing data"

The paper's Eq. 2 replaces S by (1−m)S. The older view shrinks the sample, to N(1−m) (the paper's Fig. 2c). The third curve, N(1−m)², is the sample-size version that has the same threshold as Eq. 2.

Eq 2: S→(1−m)SN(1−m)N(1−m)²no missing data

2. Run the experiment

Draws a fresh data set in your browser (D = 150), deletes entries at random, fills the gaps with zero and finds the top principal direction by power iteration. The paper's EM probabilistic PCA lands on the same curve; see the notes.

Press the button. Each dot is one random data set, so expect scatter of a few hundredths, larger near the transition (D is small here).

Theory: R² = 0 if αS′² < 1, else (αS′² − 1)/(S′ + αS′²), with S′ = (1−m)S. Critical missing rate mc = 1 − 1/(S√α).