Take-home E — low rank does not mean clean

Cut a real voice recording into short overlapping windows, take the STFT, truncate its SVD, invert, and measure SNR against the clean reference. 4.949 seconds at 48 kHz gives a (513, 465) STFT, so full rank is 465. The noisy input measures 5.00 dB, as constructed.

k	SNR	Retained energy
2	2.29 dB	33.2%	worse than the noise we started from
5	4.76 dB	51.1%	still worse
20	8.48 dB	70.1%
40	9.08 dB	78.3%	best — +4.08 dB, at 8.6% of full rank
80	7.68 dB	87.1%	past the peak
465	5.00 dB	100%	exactly the noisy input again

Both ends fail, for opposite reasons. And note: going from k=40 to k=80 retains more energy and produces worse audio, because the energy being added back is noise.

Retained energy is not the criterion. Low rank does not mean clean; it means small.