Advanced 12 min 4 of 4

Why not just regress?

A regressor returns the mean of the plausible answers. A sampler returns one of them — and for an ill-posed inverse problem those differ in a way you can name.

What this repairs

Not being able to say what a diffusion dehazer gives you that an L2 regressor does not — which is the question that decides whether any of this was worth it.

Assumes

One encoder, two decoders
Why 25 steps and not 1000 · Diffusion
What the network actually predicts · Monocular Depth

Everything to this point is machinery. A UNet that maps a hazy image to a clear one, trained with an $L2$ loss, is one forward pass, one loss term and no schedule, no sampler, no codec convention and no prediction-type footgun. It is enormously simpler and it should be the default answer.

So the question that decides whether the last three topics were worth building is: what does sampling give you that the regressor does not? The answer is not "sharper images", although that is what people say. It is a statement about what the two objectives converge to, and it is short enough to derive.

What an L2 regressor converges to

$f^{*} = \arg\min_{f}\ \mathbb{E}_{(x, y)}\left[\left\lVert f(y) - x \right\rVert^{2}\right]$

The training objective of the regressor, over the joint distribution of hazy inputs y and clear targets x. Note it is an expectation over pairs — the network never sees a single answer, it sees a distribution of them.
$= \arg\min_{f}\ \mathbb{E}_{y}\left[\ \mathbb{E}_{x \mid y}\left[\left\lVert f(y) - x \right\rVert^{2}\right]\ \right]$

Split the expectation by conditioning on y. The inner problem is now independent for each input, so the optimum can be found pointwise — this is the step that turns a statement about training into a statement about a single input.
$f^{*}(y) = \mathbb{E}\left[x \mid y\right]$

Minimising a squared error over a distribution returns its mean, which is the standard result and the entire answer. An L2 regressor does not learn to produce a plausible clear image; it learns to produce the average of every clear image consistent with its input.
$p(x \mid y) \text{ broad } \;\Longrightarrow\; \mathbb{E}[x \mid y] \notin \operatorname{supp} p(x \mid y)$

And an average of plausible images is generally not a plausible image. Where the posterior is wide the mean falls off the data manifold entirely — this is why regressed outputs blur, and blur is not a capacity problem that a larger network fixes.
$x \sim p_\theta(x \mid y)$

A sampler does something categorically different: it returns a draw. The output is on the manifold by construction, and it is not the mean of anything — which is the property being bought, at the cost of it no longer being the estimate that minimises squared error.

Insight Where the posterior is actually wide

For dehazing, the posterior width is not uniform over the image and you can say exactly where it grows.

The observation model from the fog topic is a mix between the scene radiance and the airlight, weighted by transmission. Where transmission is high — near field, thin haze — the clear image is nearly recoverable and $p(x \mid y)$ is narrow: a regressor and a sampler will agree, and the regressor is the better-behaved estimate. Where transmission is low — far field, thick haze — the observation is almost entirely airlight and the scene contributes a few code values above the noise floor. The posterior there is wide, and it is wide for a physical reason rather than because the model is uncertain.

That is the regime the whole apparatus is for. In it, the regressor returns a grey smear because a grey smear is genuinely the mean of the possibilities, and the sampler returns one sharp guess. The sharp guess is not more accurate. Per pixel it may well be further from the truth. It is on the manifold, and whether that is worth anything depends entirely on what consumes it.

Pitfall The metric will go the wrong way, and it is right to

PSNR, MSE and every per-pixel distortion metric are minimised by the conditional mean. That is not a coincidence — it is the same identity as the derivation above, read backwards.

So a diffusion dehazer will usually score worse on PSNR than an $L2$ regressor trained on the same data, while looking better to a person. Neither number is wrong. This is the perception–distortion tradeoff, and it is a statement about the two objectives being genuinely different rather than about one model being under-trained.

The concrete failure is reporting one and claiming the other. A paper that reports PSNR and shows crops has reported the axis on which its method loses and illustrated the axis on which it wins. If the deliverable is a metric, the regressor may simply be the correct engineering answer — and this whole arc is over-engineering, which is a defensible thing to conclude and should be concluded out loud rather than avoided.

For the depth half the argument is subtler, and it lands on a question the monocular-depth topic already opened: what a depth prediction even means.

A monocular depth network's output is already defined only up to a transformation, and its metrics are computed after an alignment the robot cannot perform at inference time. Adding a sampler does not fix that — nothing on this page fixes that. What it adds is a second, orthogonal thing: run the sampler twice with different seeds and you get two depth maps, both plausible, differing most where the image is least informative. That spread is a posterior width, measured rather than asserted, and it is available for free from a model you were going to run anyway.

A regressor cannot offer it at all. It produces one number per pixel and no way to ask how sure it was; any uncertainty it reports has to be a second head trained on a second objective, which is a separate model with separate failure modes.

Be precise about the status of this, though. The shipped configuration has test-time depth ensembling present but disabled — the ensemble size is one, with a median reduction and a regulariser weight waiting for it. The mechanism exists, it is the right mechanism, and nobody has turned it on. That is the honest state of it, and it is a more useful thing to know than a claim that the model quantifies its own uncertainty.

In practice The bill, itemised

Against a single forward pass, what this architecture costs:

Twenty-five forward passes for RGB and twenty for depth, per image, per task, in the default mode — and in that mode each pass runs the shared encoder again.
A frozen codec between the model and the pixels, with a reconstruction ceiling nothing downstream can lift.
Three silent conventions — the latent scaling factor, the prediction type, and the schedule's terminal-SNR handling — each of which produces a plausible wrong image when it is wrong rather than an error.
A reproducibility story that does not survive a change of joint mode.

Set against: samples that lie on the data manifold where the posterior is wide, and a free uncertainty estimate from repeated draws.

That is a real trade with a real answer, and the answer depends on the regime. For a well-lit scene in thin haze, evaluated by a distortion metric, the regressor wins and should be used. The case for everything in these three topics is the ill-posed end — thick haze, far field, and a consumer that needs a plausible answer rather than an average one.

Check yourself

Your diffusion dehazer scores 1.5 dB worse PSNR than the L2 baseline you are replacing. Do you ship it?

Show answer

Not on that number alone, in either direction — and there is a specific diagnostic that turns the question into a measurement.

Run the sampler several times on the same inputs and look at where the draws disagree. If they disagree in exactly the regions where the baseline is smoothest, you are looking at posterior width, and the $1.5$ dB is the expected price of not averaging. That is the intended behaviour, not a regression, and the PSNR gap is evidence for the model rather than against it.

If the draws agree everywhere and the model is simply worse, the gap is not buying anything and you have a genuinely worse model — check the prediction type, the scaling factor and the terminal-SNR handling before anything else, because all three produce plausible output when wrong.

Then answer the actual question, which is who consumes the output. A downstream detector or a human wants the sample. A distortion metric wants the mean. If the deliverable is the metric, ship the regressor and say why.