Training losses: design notes

Why the losses in pyFDN.train.losses are shaped the way they are. The reasoning is qualitative on purpose: what is stated here is a design choice and its rationale, not a benchmark. Behaviour is FDN- and seed-dependent, so measure your own case before assuming any of it transfers.

Flatness on the training grid depends on nfft

FlatMagnitude measures |H| at the model’s DFT bins, so its frequency resolution is the model’s nfft. The truncation to nfft samples is a rectangular window, so the peak-to-median range of |H| grows with nfft, and the mean-squared error weights the tallest modes ever more heavily. Loss values are therefore comparable across runs at one nfft and not across different ones. A multi-resolution loss (FlatSpectrogram), whose analysis windows set their own resolution, is the way out.

The optimization crosses long plateaus on FlatMagnitude; train_fdn’s default patience=10 stops inside one. Raise it for a converged fit.

Asymmetric flatness: peaks over dips

AsymmetricFlatMagnitude raises the two sides of the deviation from flat to different powers so peaks cost more than dips. The exponent, not a weight, is what makes it bite: at peak_power=4 a peak twice as tall costs sixteen times as much. Flat stays the unique minimum at every peak_power (normalizing by RMS forces <(1 + d)^2> = 1), and the loss is gain-invariant.

It is deliberately not in decibels: ∂dB/∂|H| ∝ 1/|H|, so in dB the deepest nulls dominate the gradient however lightly weighted. In linear magnitude the gradient is ∝ p·(d⁺)^(p-1), largest at the tallest peaks, while a dip is bounded at d = -1.

The cost is steps and steadiness: that same gradient vanishes near the optimum, so a higher exponent takes longer to converge and varies more from seed to seed. The advantage over FlatMagnitude is not unconditional. Loss values are not comparable across peak_power, or with FlatMagnitude; compare the responses.

Multi-resolution flatness: average over frames first

FlatSpectrogram averages the short-time magnitudes over frames into a Welch estimate before measuring flatness. Asking each individual frame to be flat instead is actively harmful: an isolated echo inside a short frame already has a flat frame spectrum, so that objective rewards an impulsive, comb-filtered IR.

Energy decay vs. spectrogram distance

MatchEnergyDecay compares octave-band Schroeder curves. A magnitude spectrogram distance is not a substitute for fitting a decay: it compares two signals frame by frame, and two rooms with identical decay have uncorrelated fine structure, so predicting silence scores better there than predicting the right amount of the wrong detail. Fitted to a measured room, a spectrogram distance is minimized by an FDN whose RT falls well short of the measurement, where an energy-decay loss is not.

Doubly-cumulated energy

MatchCumulativeEnergy integrates the short-time power spectrum twice — backwards in time and along frequency — and scores the RMS difference of the two surfaces after a compressive power.

Why cumulate twice. The time direction is Schroeder backward integration (the reason MatchEnergyDecay exists). The frequency direction replaces splitting into octave bands: band edges are an arbitrary quantization a fit can satisfy on average while getting the shape wrong, whereas a cumulative sum is the limit of ever-finer bands and is monotone and smooth in both axes, which a gradient values. Read down the t = 0 edge and you have the integrated spectrum (colour); read across the f = 0 edge and you have the full-band decay; the interior ties them together band by band.

Compression, not decibels. The surface spans the whole dynamic range of the decay, so a plain MSE would see only the first frames. Raising the normalized surface to a fractional power (0.5 default) compresses that range while staying bounded, its gradient ∝ x^(p-1) finite everywhere the floor allows. A logarithm is worse: it turns the silence below the response into an unbounded penalty dominated by whichever bin is nearest zero.

Which way the frequency cumulation runs. Cumulating downwards alone — the default, and the plain reading of “energy above this frequency” — leaves the bottom octave with almost no gradient, and a fit that has to find a decay tends to abandon it there. frequency="both" scores both directions and averages, which recovers the low end; it is the setting to reach for when the decay is not given. This is a matter of weighting rather than frequency resolution, so a longer analysis window is not the fix.