Diffusion Models

Definition

Diffusion Model

A diffusion model is a generative model that learns a data distribution $p({\mathbf{x}}_0)$ by reversing a fixed noising process. A forward (diffusion) process gradually corrupts a data point ${\mathbf{x}}_0$ into pure Gaussian noise ${\mathbf{x}}_T$ over $T$ steps; a learned reverse (denoising) process $p_{\theta }({\mathbf{x}}_{t-1}\mid {\mathbf{x}}_t)$ removes noise step-by-step to sample new data from noise. In recommendation it is the non-LLM generative backbone of slide RS-L03b’s “Generative Model (e.g., LLM, Diffusion)” box: it can either denoise recommender embeddings back onto the existing item pool, or generate new item content.

Intuition

Learn to denoise, then run the camera backwards

Imagine slowly adding static to a photo until nothing is left but noise — that is the fixed forward process, no learning required. The model’s only job is to learn the reverse step: given a noisy image at level $t$, predict the noise that was added, and subtract a bit of it. Stack $T$ such tiny denoising steps and you can start from pure noise and walk back to a clean sample. Because each step solves an easy regression problem (predict the noise), training is stable — unlike a GAN’s adversarial game. For recommendation, “the photo” can be a user/item embedding (denoise it toward a real catalogue item, conditioned on collaborative signal) or actual content like a fashion image.

Mathematical Formulation

Forward noising, reverse denoising, and the training loss (DDPM)

Forward process (fixed, no parameters) — add Gaussian noise on a variance schedule ${\beta }_1,\dots ,{\beta }_T$: $q({\mathbf{x}}_t\mid {\mathbf{x}}_{t-1})=\mathcal{N}({\mathbf{x}}_t;\sqrt{1-{\beta }_t}{\mathbf{x}}_{t-1},{\beta }_t\mathbf{I})$ A convenient closed form lets us jump to any step $t$ in one shot (with ${\alpha }_t=1-{\beta }_t$, ${\bar{\alpha }}_t={\prod }_{s=1}^t{\alpha }_s$): ${\mathbf{x}}_t=\sqrt{{\bar{\alpha }}_t}{\mathbf{x}}_0+\sqrt{1-{\bar{\alpha }}_t}\mathbf{ϵ},\mathbf{ϵ}\sim \mathcal{N}(0,\mathbf{I})$ Reverse process (learned) — a network parameterizes each denoising step: $p_{\theta }({\mathbf{x}}_{t-1}\mid {\mathbf{x}}_t)=\mathcal{N}({\mathbf{x}}_{t-1};{\mathbf{\mu }}_{\theta }({\mathbf{x}}_t,t),{\Sigma }_{\theta }({\mathbf{x}}_t,t))$ Training objective — instead of the full variational bound, DDPM trains a noise-predictor ${\mathbf{ϵ}}_{\theta }$ with a simple MSE: ${\mathcal{L}}_{\text{simple}}={\mathbb{E}}_{{\mathbf{x}}_0,\mathbf{ϵ},t}[\Vert \mathbf{ϵ}-{\mathbf{ϵ}}_{\theta }(\sqrt{{\bar{\alpha }}_t}{\mathbf{x}}_0+\sqrt{1-{\bar{\alpha }}_t}\mathbf{ϵ},t\left){\Vert }^2\right]$

where:

• ${\mathbf{x}}_0$ — clean data sample (an image, or a recommender embedding)
• ${\mathbf{x}}_t$ — the sample after $t$ noising steps; ${\mathbf{x}}_T\approx \mathcal{N}(0,\mathbf{I})$
• ${\beta }_t\in (0,1)$ — variance schedule (how much noise is added at step $t$)
• ${\alpha }_t=1-{\beta }_t$, ${\bar{\alpha }}_t={\prod }_{s\le t}{\alpha }_s$ — cumulative signal retained up to step $t$
• $\mathbf{ϵ}$ — the Gaussian noise actually added; ${\mathbf{ϵ}}_{\theta }$ — the network’s prediction of it
• $t$ — diffusion step, sampled uniformly from $\{1,\dots ,T\}$ during training
• ${\mathbf{\mu }}_{\theta },{\Sigma }_{\theta }$ — mean/covariance of the learned reverse step (recoverable from ${\mathbf{ϵ}}_{\theta }$)

Conditioning. To make sampling controllable (the recommendation use), the denoiser takes a condition $\mathbf{c}$ — e.g., user history or a CF embedding — giving ${\mathbf{ϵ}}_{\theta }({\mathbf{x}}_t,t,\mathbf{c})$, optionally combined with Classifier-Free Guidance.

Key Properties / Variants

• Sampling (ancestral) algorithm. Generation is the reverse loop, one denoising step at a time:

Algorithm: DDPM Sampling (conditioned on c)
──────────────────────────────────────────────
x_T ~ N(0, I)                       # start from pure noise
for t = T, T-1, ..., 1:
    z ~ N(0, I) if t > 1 else z = 0
    eps = eps_θ(x_t, t, c)          # predict the noise (c = user/CF condition)
    # one reverse step: subtract a scaled portion of predicted noise
    x_{t-1} = (1/√α_t) * ( x_t - (β_t / √(1-ᾱ_t)) * eps ) + √β_t * z
return x_0                          # clean sample (embedding or content)

• Why it is stable to train. The loss is a plain per-step MSE on noise — no adversarial min-max (unlike GANs), no autoregressive token ordering. This is the key contrast with the autoregressive semantic-ID decoders (TIGER-style) that dominate the rest of the GenRec lecture.
• Three roles “generative” plays in RecSys (RS-L04 slide 3 explicitly disambiguates the term):
  1. Generate item identifiers — autoregressive over semantic IDs (TIGER); not a diffusion model. This is the main lecture focus.
  2. Diffusion for embedding denoising (DDRM, SIGIR 2024) — diffusion denoises user/item embeddings; collaborative signal conditions the reverse process; output is grounded in the existing item pool, so no new item content is created.
  3. Diffusion for content generation (DiFashion, SIGIR 2024) — generates new item content (fashion images) conditioned on user history + constraints.
• Trainable or frozen. On RS-L03b’s generative-recommender diagram the backbone carries both flame and snowflake icons — a diffusion denoiser can be trained on platform data or used as a frozen pretrained generator.
• Continuous vs. discrete output. Diffusion operates naturally on continuous vectors (embeddings, pixels). To recommend a real item it must be grounded: either denoise toward and look up the nearest catalogue embedding (DDRM), or pair with a retrieval/ranking step — analogous to the validity/grounding problem the autoregressive route solves with a trie.
• Latent diffusion. Running the process in a compressed latent space (rather than raw pixels/full embeddings) cuts cost — the standard trick for image generators and applicable to large recommender embedding spaces.
• Cost. Sampling needs many sequential reverse steps ($T$ can be hundreds), so inference latency is a real concern under a recommendation serving budget, mirroring the decoding-cost limitation of generative recommenders generally.

Connections

• Sibling generative backbone to the autoregressive semantic-ID route in Generative Recommendation (the “LLM, Diffusion” alternatives)
• Controllable sampling via Classifier-Free Guidance
• Contrast with adversarial / token-by-token generators; aligns with Next-Item Prediction when conditioned on a user history
• Used as a planner in RL via the Decision Diffuser / decision diffusion line (sequence generation over trajectories)
• Embedding-denoising variant grounds output using ideas from Dense Retrieval (nearest-item lookup)
• Quantization-based alternative for discrete item codes: RQ-VAE / Semantic IDs

Appears In

• RS-L03b - From LLMs to LRMs
• RS-L04 - Generative Recommendation