# NoisyDataFidelity

### *class* deepinv.sampling.NoisyDataFidelity(weight=1.0, \*args, \*\*kwargs)

Bases: [`DataFidelity`](https://deepinv.org/api/stubs/deepinv.optim.DataFidelity.html.md#deepinv.optim.DataFidelity)

Data fidelity term for noisy input data $- \log p(y|x + \sigma(t) \omega)$ with $\omega\sim\mathcal{N}(0,\mathrm{I})$.

This is a base class used for approximating $- \log p_t(y|x_t)$ used in diffusion
algorithms for inverse problems, in [`deepinv.sampling.PosteriorDiffusion`](https://deepinv.org/api/stubs/deepinv.sampling.PosteriorDiffusion.html.md#deepinv.sampling.PosteriorDiffusion).

It comes with a `.grad` method computing the negative log-likelihood gradient :math:\` - nabla_{x_t} log p_t(y|x_t)\`.

By default we have

$$
- \nabla_{x_t} \log p(y|x + \sigma(t) \omega) = A^\top(\forw{x_t}-y),
$$

which subclasses override with their own approximation.

#### NOTE
Unlike [`deepinv.optim.DataFidelity`](https://deepinv.org/api/stubs/deepinv.optim.DataFidelity.html.md#deepinv.optim.DataFidelity), these terms are defined through their
gradient only: the approximations of $- \log p_t(y|x_t)$ used by diffusion
samplers generally do not admit a closed-form value. The `.forward` method
therefore raises a `NotImplementedError` unless a subclass defines it.

* **Parameters:**
  **weight** ([*float*](https://docs.python.org/3.12/library/functions.html#float)) – Weighting factor for the data fidelity term. Default to 1.

#### forward(\*args, \*\*kwargs)

Not implemented: noisy data-fidelity terms are defined through their gradient
$-\nabla_{x_t} \log p_t(y|x_t)$, see [`grad()`](#deepinv.sampling.NoisyDataFidelity.grad). Subclasses whose
approximation of $- \log p_t(y|x_t)$ has a closed form override this method.

#### grad(x, y, physics, \*args, \*\*kwargs)

Computes the gradient of the data-fidelity term $\lambda A^\top(A(x) - y)$.

* **Parameters:**
  * **x** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – Current iterate.
  * **y** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – Input data.
  * **physics** ([*deepinv.physics.Physics*](https://deepinv.org/api/stubs/deepinv.physics.Physics.html.md#deepinv.physics.Physics)) – physics model
* **Returns:**
  (torch.Tensor) data-fidelity term.
* **Return type:**
  [*Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)

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## Examples using `NoisyDataFidelity`:

<div class="sphx-glr-thumbnails">
<!-- thumbnail-parent-div-open --><div class="sphx-glr-thumbcontainer" tooltip="This demo shows you how to use our wrapper deepinv.models.DiffusersDenoiserWrapper to turn any SOTA models from the HuggingFace Hub to an image denoiser. It also can be used to perform unconditional image generation or for posterior sampling.">![](auto_examples/sampling/images/thumb/sphx_glr_demo_diffusers_thumb.png)

[Using state-of-the-art diffusion models from HuggingFace Diffusers with DeepInverse](https://deepinv.org/auto_examples/sampling/demo_diffusers.html.md)

  <div class="sphx-glr-thumbnail-title">Using state-of-the-art diffusion models from HuggingFace Diffusers with DeepInverse</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This demo shows you how to use deepinv.sampling.PosteriorDiffusion to perform posterior sampling. It also can be used to perform unconditional image generation with arbitrary denoisers, if the data fidelity term is not specified.">![](auto_examples/sampling/images/thumb/sphx_glr_demo_diffusion_sde_thumb.png)

[Building your diffusion posterior sampling method using SDEs](https://deepinv.org/auto_examples/sampling/demo_diffusion_sde.html.md)

  <div class="sphx-glr-thumbnail-title">Building your diffusion posterior sampling method using SDEs</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="In this tutorial, we will go over the steps in the Diffusion Posterior Sampling (DPS) algorithm introduced in chung2022diffusion. The full algorithm is implemented in deepinv.sampling.DPS.">![](auto_examples/sampling/images/thumb/sphx_glr_demo_dps_thumb.png)

[DPS – Posterior Sampling with Diffusion Models](https://deepinv.org/auto_examples/sampling/demo_dps.html.md)

  <div class="sphx-glr-thumbnail-title">DPS -- Posterior Sampling with Diffusion Models</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This demo shows you how to perform unconditional image generation and posterior sampling using Flow Matching (FM).">![](auto_examples/sampling/images/thumb/sphx_glr_demo_flow_matching_thumb.png)

[Flow-Matching for posterior sampling and unconditional generation](https://deepinv.org/auto_examples/sampling/demo_flow_matching.html.md)

  <div class="sphx-glr-thumbnail-title">Flow-Matching for posterior sampling and unconditional generation</div>
</div><div class="sphx-glr-thumbcontainer" tooltip="This example compares six approximations of the measurement-matching term used by diffusion posterior samplers.">![](auto_examples/sampling/images/thumb/sphx_glr_demo_noisy_data_fidelity_thumb.png)

[Noisy data-fidelity terms for diffusion posterior sampling](https://deepinv.org/auto_examples/sampling/demo_noisy_data_fidelity.html.md)

  <div class="sphx-glr-thumbnail-title">Noisy data-fidelity terms for diffusion posterior sampling</div>
</div>
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