Executive Summary

A diffusion model works in two phases. In the forward process, noise is added to clean data in small increments until it becomes pure noise; in the reverse process, that noise is removed step by step to recover the data. Training means learning, by regression, a function $x_\theta(x_t, t)$ that takes a noisy sample $x_t$ at each time $t$ and produces the most plausible estimate of the original data $x_0$. The objective is the expected reconstruction error.

Eq. 1. The diffusion denoising objective — a regression that predicts the original from a noisy sample.

The optimal denoiser is the posterior mean $\mathbb{E}[x_0 \mid x_t]$, which by Tweedie's formula corresponds one-to-one with the score function $\nabla \log p_t(x)$. So "denoising well = estimating the score well = knowing the data distribution" all collapse into one statement. The trouble is the cost of learning that distribution from scratch. With no assumption about structure, estimating the score of an $n$-dimensional distribution to $\epsilon$ accuracy requires a number of samples that explodes exponentially in the dimension $n$, as $O(\epsilon^{-n})$. In the paper's words, "$\epsilon$-accurate score estimation requires $O(\epsilon^{-n})$ training samples."

1.1The Root of the Curse — the Minimax Limit of Nonparametric Estimation

This exponential blow-up is not a weakness peculiar to diffusion; it comes from a long-standing limit in statistics. When you estimate an $s$-smooth function in $d$ dimensions, the best achievable error decays with the number of samples $N$ only as $N^{-s/(2s+d)}$. To hit a target accuracy $\epsilon$ you need roughly $N \sim \epsilon^{-(2s+d)/s}$ samples, and because the dimension $d$ sits in the exponent's denominator, the sample requirement explodes as the dimension grows. The intuition is a grid: split each axis into 100 cells and a $d=10$ dimensional space already has $10^{20}$ cells. Filling all of them with samples to trace out the distribution is effectively impossible.

The numbers make it starker. Fix the smoothness at $s=2$ and the target accuracy at $\epsilon=0.01$, then grow only the dimension $d$, and you see how uncontrollably the sample requirement balloons. Merely doubling the dimension from 8 to 16 multiplies the requirement by roughly ten thousand.

But the data we actually use isn't 16-dimensional. The ambient dimension of one ImageNet photo is 150,528. Apply the table above literally and a diffusion model would have to demand more samples than there are atoms in the universe. Yet real diffusion models work just fine on a few million images. That contradiction is the paper's starting point. The answer is simple — data actually leaves that enormous space almost entirely empty and clusters in a far narrower region.

A 224×224 image with randomly colored pixels is almost always meaningless noise. Natural photographs occupy only a vanishingly small region of the full pixel space. The directions along which a face photo varies are a limited set of factors — expression, angle, lighting, hairstyle — and the number of such factors is overwhelmingly smaller than the number of pixels. This is the manifold hypothesis: high-dimensional data concentrates near a low-dimensional manifold (a curved surface) inside the high-dimensional space. It is one of the pillars of modern machine learning, hardened by Tenenbaum et al.'s Isomap (2000) and by Fefferman, Mitter, and Narayanan's statistical formalization (2016).

The "number of directions of variation" is quantified by the intrinsic dimension (ID). Several estimation studies report that the ID of standard image datasets is two to three orders of magnitude smaller than the pixel count. The table below shows representative estimates. Because the same dataset can yield very different values depending on the estimation method (MLE / TwoNN / GeoMLE) and the neighborhood size $k$, every figure should be read as an estimate.

Table 1. Ambient vs. intrinsic dimension estimates for image datasets — per Pope et al. (ICLR 2021). All values are estimates that depend on method and neighborhood size.

The paper takes three empirical observations as its starting point. First, the low intrinsic dimension of images. Second, that images lie not on a single manifold but on a union of multiple manifolds, one per class or mode (Brown et al., ICLR 2023). Third, that the denoising autoencoder inside a trained diffusion model is empirically low-rank. These three observations underpin the data model in the next section. The second is the crux — data is split into several surface patches rather than one curved surface, and each patch can be locally approximated by a flat plane, i.e., a low-dimensional subspace.

To translate the intuition into mathematics, you need a tractable data model. The paper assumes data is drawn from a mixture of $K$ low-rank Gaussians (mixture of low-rank Gaussians, MoLRG hereafter). Component $k$ has mean $\mu_k^\star$ and a covariance of rank $d_k < n$. Equivalently, a sample from component $k$ is generated as follows.

Eq. 2. Generating a sample from MoLRG component k — a latent variable $z$ mapped through the low-dimensional subspace basis $U_k^\star$ onto the mean $\mu_k^\star$.

Here $n$ is the ambient dimension (pixel count), and $U_k^\star$ is an $n\times d_k$ column-orthogonal matrix — the basis of the low-dimensional subspace that component $k$ lives on. $d_k$ is the dimension of that subspace, i.e., the intrinsic dimension of component $k$. The intrinsic dimension of the whole dataset is taken to be $d = \max_k d_k$. Pictorially, each component is a single flat plane sitting tilted inside the high-dimensional space, and the data is scattered near $K$ such planes.

This model is reasonable for two reasons. First, it matches the empirical finding that the ID of images is overwhelmingly smaller than the ambient dimension ($d_k \ll n$). Second, locally linearizing a "union of manifolds" gives exactly this form. MoLRG is, in other words, a statistically tractable translation of "a union of locally flat, low-dimensional manifolds." Simple as it is, it captures the essential structure — low-dimensionality and multimodality — and so becomes the minimal stage on which the essence of diffusion training can be analyzed.

Here is the paper's first central result. Under the MoLRG assumption, training a diffusion model is exactly equivalent to the canonical subspace clustering problem from classical statistics. Subspace clustering is the classical problem of looking at points scattered across several low-dimensional subspaces and simultaneously (1) clustering each point into the subspace it belongs to and (2) finding the orientation (basis) of each subspace. It is precisely the problem that GPCA and SSC, in the lineage of Yi Ma, have been solving for decades.

4.1A Theoretically Principled Parameterization

For the equivalence to hold, the diffusion network can't be set up arbitrarily. Starting from the fact that under MoLRG the optimal denoiser of each component is a "shrunken orthogonal projection onto the corresponding subspace," the paper derives the following parameterization.

Eq. 3. The diffusion parameterization derived from the MoLRG optimal denoiser — $U_k$ is the subspace basis to be learned.

Here $U_k$ is the subspace basis training must find, $D_k$ is the shrinkage diagonal matrix set by the noise level, $w_k(x_t)$ is the soft responsibility for which component the sample belongs to, and $s_t$ is the noise scale. The point is that what training effectively searches for is $U_k$ — the orientation of the subspace each data mode lives on. This parameterization is not chosen arbitrarily; it is derived from the optimal denoiser, and it is consistent with the third empirical observation from the previous section (the low-rank nature of the denoiser).

4.2The Equivalence Theorem

Substituting this parameterization into the reconstruction objective of Eq. 1 and working through the Gaussian integral, minimizing the diffusion training objective becomes equivalent to the following maximization problem (the equivalence theorem, in the family of the paper's Theorem 3).

Eq. 4. The equivalent objective — assign each sample to the subspace where its projection energy is maximal, and maximize the sum of that energy (= canonical subspace clustering).

Read it this way. Assign each sample $x^{(i)}$ to the subspace $C_k$ where its projection energy $\lVert U_k^\top x \rVert^2$ is largest, and find the basis $U_k$ that maximizes the total projection energy of the samples so assigned. That is exactly the definition of K-subspace / subspace clustering. Put differently, diffusion reconstructing data well is the same thing as it having found the precise subspace each data mode lives on. Reconstruction and clustering become two sides of the same coin.

4.3The Proof Skeleton

The proof boils down to four steps. (1) Show that under MoLRG the optimal denoiser of each component is a shrunken projection onto the subspace (Tweedie's formula + Gaussian integral → the form of $D_k$). (2) Substituting Eq. 3 into the regression loss and tidying the Gaussian integral turns the loss into "$-\sum \lVert U_k^\top x \rVert^2 +$ const," so minimizing the loss flips into maximizing projection energy. (3) In a particular signal-to-noise-ratio (SNR) regime, the soft responsibilities $w_k$ converge to hard assignments, so each sample is assigned to its nearest subspace and the clusters $C_k$ are well defined. (4) As a result, the training landscape of diffusion coincides with that of subspace clustering. The conclusion: optimizing diffusion is the same as solving the classical problem GPCA and SSC have long solved.

If diffusion training is equivalent to subspace clustering, you can import subspace clustering's recovery theory wholesale. This is the paper's second central result and the most practically powerful part — sample complexity is determined by the intrinsic dimension, not the ambient dimension.

5.1Single-Subspace Recovery — a Sharp Phase Transition

Start with the simplest case of a single component ($K=1$) (the single-Gaussian recovery theorem, in the family of the paper's Theorem 2). If the number of samples is at least the subspace dimension ($N \ge d$), the subspace can be recovered exactly when noise is small, and the estimation error shrinks as samples grow. Conversely, if there are fewer samples than the dimension ($N < d$), recovery is information-theoretically impossible — the samples simply don't carry enough information to pin down the subspace. Recovery error is measured by the Frobenius distance between the projection matrices of the estimated and true subspaces.

Eq. 5. Upper bound on single-subspace recovery error — the denominator's $\sqrt{N}-\sqrt{d-1}$ creates the $N\approx d$ threshold ($c_1$ is an absolute constant).

Look at the denominator of Eq. 5. As $N$ approaches $d$, $\sqrt{N}-\sqrt{d-1}$ goes to zero and the error bound blows up; as $N$ exceeds $d$, the error rapidly stabilizes. That denominator is the mathematical identity of the sharp phase transition at $N \approx d$. What matters is that this threshold is entirely independent of the ambient dimension $n$. Whether there are 48 pixels or 150,000, the samples needed to recover the subspace depend only on the intrinsic dimension $d$. The constant $c_1$ is stated as an absolute constant independent of the data and dimension, but its exact value is not specified (a qualitative guarantee).

5.2Mixture Recovery — Extending to K Components

The conclusion holds for the general MoLRG with $K$ components as well (the mixture recovery theorem, in the family of the paper's Theorem 4). If each component has $N_k \ge d$ samples, every subspace can be recovered with the same error bound, and the success probability approaches 1 exponentially as samples grow. The total sample complexity is roughly $N \approx K \cdot d$ — linear in the intrinsic dimension $d$, roughly linear in the number of components $K$, and non-exponential in the ambient dimension $n$. In one line: sample complexity is on the order of $\tilde{O}(K \cdot d)$ and independent of pixel count.

Where parallel theory (e.g., Gatmiry et al.) handles complexity via the total-variation (TV) distance between distributions, this paper's strength is that it pins down an explicit phase transition for subspace recovery. By drawing the "works or doesn't" boundary precisely in terms of sample count, it becomes more than theory — it becomes a prediction testable by experiment.

The $N \approx d$ phase transition that the theory predicts is observed directly in experiments. The paper confirms it on two stages: controlled synthetic data, and diffusion models trained on real images.

6.1Synthetic Experiments — the Critical Line Is Visible

The synthetic experiments fix the ambient dimension at $n=48$ and vary the intrinsic dimension $d$ over 2–8, the sample count $N$ over 2–15, and the number of components $K$ over 1–3, repeating each setting 20 times (per the paper's synthetic experiments). When you color whether subspace recovery succeeded onto a grid of sample count against intrinsic dimension, the failure and success regions split cleanly along the $N \approx d$ diagonal. Crucially, this boundary appears solely at $d$ with no relation to the ambient dimension of 48 — exactly as the theory predicts.

6.2Real Images — the Abrupt Shift from Memorization to Generalization

Train a U-Net diffusion model on real images (MNIST, CIFAR-10, CelebA/FFHQ, etc.) and the theory's phase transition shows up as an abrupt shift in generalization. With little training data, the model essentially memorizes the training samples (memorization). But the moment the number of training samples crosses a threshold proportional to the intrinsic dimension, the model begins producing new images it never saw in training (generalization). The paper quantifies the degree of generalization with a metric $GL$ measured via self-supervised copy-detection features — if the generations differ enough from the training samples, $GL \to 1$ (generalization); if they nearly copy, $GL \to 0$ (memorization).

The observation is clear. The number of samples at which generalization begins scales linearly with the dataset's intrinsic dimension. The "recovery succeeds the moment $N$ exceeds $d$" seen in the synthetic experiments reappears on real data as "generalization begins the moment $N$ crosses a threshold proportional to the intrinsic dimension." With controlled mathematics and messy real data pointing to the same boundary, this phase transition is not an artifact but a real phenomenon created by the low-dimensional structure of data.

The subspace basis $U_k$ that diffusion discovers is not just a mathematical direction. The paper shows that this basis aligns with human-readable semantic attributes. Analyzing the principal components of the trained denoiser's Jacobian — i.e., the basis directions of the discovered subspace — on FFHQ face data, each direction corresponds to a semantic attribute such as gender, hairstyle, or color. Moving along one axis of the subspace changes only that attribute.

This provides the mathematical foundation for controllable generation. If a subspace basis is itself a semantic axis, then on top of a pretrained diffusion model you can edit an attribute with no additional training, simply by shifting the latent variable along a particular basis direction. Why an operation like "change only this face's hairstyle and keep everything else" works is explained in a single sentence: subspace alignment.

To place this paper, you have to look at three research streams together: the manifold hypothesis and intrinsic dimension; the theory of low-dimensional learning in diffusion; and subspace clustering in the lineage of Yi Ma. The paper's originality lies not in starting a new stream but in building a bridge that connects two separately developed streams — modern diffusion and classical subspace clustering — as an exact equivalence.

8.1The Manifold Hypothesis and Diffusion Low-Dim Theory

The manifold hypothesis was hardened by Tenenbaum et al. (Isomap, Science 2000) and Fefferman et al. (J. AMS 2016), while intrinsic-dimension estimation runs from Levina-Bickel's MLE estimator (NeurIPS 2004) through Pope et al. (ICLR 2021) to Brown et al.'s verification of the union-of-manifolds hypothesis (ICLR 2023). There were already several theories that diffusion adapts to low-dimensional data — Chen (Minshuo) et al. (ICML 2023), Oko et al.'s minimax optimality (ICML 2023), De Bortoli's manifold convergence (TMLR 2022), Shah et al.'s analysis of the DDPM objective (NeurIPS 2023), Gatmiry et al.'s Gaussian-mixture complexity, and more. This paper differentiates itself, on top of that lineage, by presenting an explicit phase transition for subspace recovery rather than a distributional distance (TV).

8.2Subspace Clustering in the Yi Ma Lineage

Subspace clustering is the lifelong theme of co-author Yi Ma. It runs through GPCA (Vidal-Ma-Sastry, TPAMI 2005), Sparse Subspace Clustering (Elhamifar-Vidal, CVPR 2009 / TPAMI 2013), and on to ReduNet (Chan et al., JMLR 2022) and MCR² (Yu et al., NeurIPS 2020). The delight of this paper is that it reduces the diffusion training objective right back to the canonical subspace clustering that GPCA and SSC have solved. Classical representation-learning theory and a modern generative model meet in a single equation — and that is what distinguishes this from "yet another diffusion theory."

8.3Score Matching and the Foundations of Diffusion

The foundational classics are worth noting too: score matching (Hyvärinen, JMLR 2005) and its connection to denoising autoencoders (Vincent, Neural Computation 2011), generation based on nonequilibrium thermodynamics (Sohl-Dickstein et al., ICML 2015), DDPM (Ho-Jain-Abbeel, NeurIPS 2020), Score-SDE (Song et al., ICLR 2021), NCSN (Song-Ermon, NeurIPS 2019), and EDM (Karras et al., NeurIPS 2022), which organized the design space. The equivalence theorem of this paper stands on the foundation they built — "denoising = score estimation."

Where these theorems touch industry is cost. The market for AI training-data licensing is projected to grow from about $4.8B in 2025 to roughly $22.6B by 2034 (18.8% CAGR), and the average license contract for an enterprise proprietary dataset rose about 34% over 2023–2025 to roughly $1.2M per deal. Video training data trades at $1–4 per minute. At the same time, training-compute costs are exploding — some forecasts put the cost of a single frontier training run at $1B around 2027. It is an environment where data and compute are getting expensive at once.

9.1From "More" to "More Structured"

The theorem that sample complexity is linear in the intrinsic dimension says that the data you need depends not on pixel count but on the "number of directions of variation." Why DreamBooth and LoRA can teach a new concept from as few as 3–15 images and a few hundred megabytes is an expression of the same principle — both presuppose the data's low-dimensional structure and quickly fit just that narrow structure. This is the macro backdrop behind the industry's shift from "more data" to "more structured data." Mass-scraped data carries legal risk too (more than 70 copyright lawsuits, with a settlement of roughly $1.5B in one case), so the strategic value of small, well-structured data only grows.

9.2Synthetic Data and Model Collapse

Gartner projects that around 2026 roughly 75% of the data used for AI will be synthetic. Yet repeatedly training on synthetic data loses diversity and collapses modes — "model collapse." Through this article's lens the essence is sharp: if the synthesis process fails to preserve the original's low-dimensional structure, the distribution gets distorted and some subspaces vanish. So the quality of synthetic data cannot be measured by statistical fidelity alone. You must also measure whether it "estimates the original's intrinsic dimension and keeps the same low-dimensional structure." The trade-off where strong differential privacy (DP) damages correlation structure (i.e., low-dimensional structure) is understood in the same vein.

9.3Structural Health as a Diagnostic Metric

Intrinsic dimension becomes a metric that quantifies a dataset's complexity and structure. Recent work that detects out-of-distribution (OOD) samples or anomalies via diffusion-based local intrinsic-dimension estimation shows that this metric can serve as a diagnostic tool. To the question "is this dataset structured enough for training," intrinsic dimension gives one quantitative answer. A diagnosis that measures the health of structure rather than the volume of data — that is the new lens this paper offers to data practice.

This paper may look like distant pure theory, but it lands right in the middle of treating data as an asset. Here are four angles on why Pebblous pays attention to this result. Each follows naturally from the paper's result to a general implication and then to a Pebblous agenda — an honest alignment, not a connection forced into place.

1Business/Technical Link — the Definition of Good Data Becomes Mathematics

That "intrinsic low-dimensional structure determines learning efficiency" has been proven as a theorem means the agendas of AI-Ready data and data quality now have theoretical backing. Good data is not abundant data but data whose structure is alive — sample complexity $N \approx d$ says exactly that in an equation. The grounds are in place to shift the center of gravity of data strategy from volume to structure.

2Data-Quality View — Structural Health as a Diagnostic Axis

Intrinsic dimension is a candidate diagnostic metric quantifying a dataset's "structural health." Set alongside research that catches anomalies and OOD via diffusion-based local intrinsic-dimension estimation, it dovetails immediately with the diagnostic philosophy of Pebblous DataClinic — reading the structure of data to diagnose its health. Beyond missingness, duplication, and label errors, it adds a diagnostic axis that asks "is this data structured enough for training."

3Practical Implication — the First Principle of Synthetic Pipelines

Preserving low-dimensional structure is the core of utility in synthetic data. Synthesis that fails to preserve structure distorts the distribution and leads to model collapse. So a synthetic-data simulation pipeline should make "estimate the original's intrinsic dimension and verify the synthetic output keeps the same structure" its first principle. Measuring structural similarity alongside — not just statistical resemblance — is what separates useful synthetic data from synthetic data that ruins models.

4Positioning — Toward Understandable, Controllable Assets

The result that subspaces align with semantic attributes (controllable generation) is the generative-model counterpart of the Data Greenhouse vision of "treating data as an understandable, controllable asset." Being able to separate and control semantic axes means seeing data not as a black lump but as a structure you can read and handle. This paper shows that this view is mathematically justified inside diffusion generative models too.

Editor's Note. Through DataClinic, which diagnoses and corrects data quality, and through synthetic-data pipelines, Pebblous has been measuring how the structure of training data affects model quality. The result this report covers — that intrinsic dimension determines learning efficiency — is the theoretical backdrop for why that work carries asset value.

Below are the sources that form the core basis of this article's account. Because the formal JMLR citation (volume, year, pages) is not yet confirmed at the time of writing, the original paper is cited only by its arXiv identifier and "accepted to JMLR" (no claims about volume or pages).

The Original Paper

• 1.Wang, P., Zhang, H., Zhang, Z., Chen, S., Ma, Y., & Qu, Q. Breaking the Curse of Dimensionality: Diffusion Models Efficiently Learn Low-Dimensional Distributions. Accepted to JMLR. arXiv:2409.02426. (v1 title: "Diffusion Models Learn Low-Dimensional Distributions via Subspace Clustering")

Intrinsic Dimension · Manifold Hypothesis

• 2.Pope, P., Zhu, C., Abdelkader, A., Goldblum, M., & Goldstein, T. (2021). The Intrinsic Dimension of Images and Its Impact on Learning. ICLR 2021. arXiv:2104.08894.
• 3.Fefferman, C., Mitter, S., & Narayanan, H. (2016). Testing the Manifold Hypothesis. Journal of the American Mathematical Society. DOI:10.1090/jams/879.
• 4.Tenenbaum, J. B., de Silva, V., & Langford, J. C. (2000). A Global Geometric Framework for Nonlinear Dimensionality Reduction (Isomap). Science. DOI:10.1126/science.290.5500.2319.
• 5.Brown, B. C. A., Caterini, A. L., Ross, B. L., Cresswell, J. C., & Loaiza-Ganem, G. (2023). Verifying the Union of Manifolds Hypothesis for Image Data. ICLR 2023.
• 6.Levina, E., & Bickel, P. J. (2005). Maximum Likelihood Estimation of Intrinsic Dimension. NeurIPS 2004.
• 7.Facco, E., d'Errico, M., Rodriguez, A., & Laio, A. (2017). Estimating the intrinsic dimension of datasets by a minimal neighborhood information. Scientific Reports 7, 12140. DOI:10.1038/s41598-017-11873-y.

Diffusion Low-Dim / Manifold Learning Theory

• 8.Chen, M., Huang, K., Zhao, T., & Wang, M. (2023). Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data. ICML 2023.
• 9.Oko, K., Akiyama, S., & Suzuki, T. (2023). Diffusion Models are Minimax Optimal Distribution Estimators. ICML 2023.
• 10.De Bortoli, V. (2022). Convergence of Denoising Diffusion Models under the Manifold Hypothesis. Transactions on Machine Learning Research (TMLR).
• 11.Shah, K., Chen, S., & Klivans, A. (2023). Learning Mixtures of Gaussians Using the DDPM Objective. NeurIPS 2023.

Subspace Clustering (Yi Ma Lineage)

• 12.Vidal, R., Ma, Y., & Sastry, S. (2005). Generalized Principal Component Analysis (GPCA). IEEE Transactions on Pattern Analysis and Machine Intelligence (TPAMI). DOI:10.1109/TPAMI.2005.244.
• 13.Elhamifar, E., & Vidal, R. (2013). Sparse Subspace Clustering: Algorithm, Theory, and Applications. IEEE TPAMI. DOI:10.1109/TPAMI.2013.57. (First presented at CVPR 2009.)
• 14.Chan, K. H. R., Yu, Y., You, C., Yang, H., Wright, J., & Ma, Y. (2022). ReduNet: A White-box Deep Network from the Principle of Maximizing Rate Reduction. JMLR 2022.
• 15.Yu, Y., Chan, K. H. R., You, C., Song, C., & Ma, Y. (2020). Learning Diverse and Discriminative Representations via the Principle of Maximal Coding Rate Reduction (MCR²). NeurIPS 2020.

Score Matching · DDPM · Nonparametric Statistics

• 16.Hyvärinen, A. (2005). Estimation of Non-Normalized Statistical Models by Score Matching. JMLR 6, 695–709.
• 17.Vincent, P. (2011). A Connection Between Score Matching and Denoising Autoencoders. Neural Computation. DOI:10.1162/NECO_a_00142.
• 18.Sohl-Dickstein, J., Weiss, E. A., Maheswaranathan, N., & Ganguli, S. (2015). Deep Unsupervised Learning using Nonequilibrium Thermodynamics. ICML 2015.
• 19.Ho, J., Jain, A., & Abbeel, P. (2020). Denoising Diffusion Probabilistic Models (DDPM). NeurIPS 2020.
• 20.Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermon, S., & Poole, B. (2021). Score-Based Generative Modeling through Stochastic Differential Equations. ICLR 2021.
• 21.Karras, T., Aittala, M., Aila, T., & Laine, S. (2022). Elucidating the Design Space of Diffusion-Based Generative Models (EDM). NeurIPS 2022.
• 22.Tsybakov, A. B. (2009). Introduction to Nonparametric Estimation. Springer. DOI:10.1007/b13794. (Standard reference for the minimax rate $N^{-2s/(2s+d)}$.)

Controllable Generation · Synthetic Data

• 23.Kwon, M., Jeong, J., & Uh, Y. (2023). Diffusion Models Already Have a Semantic Latent Space. ICLR 2023. (Unsupervised discovery of semantic directions in h-space.)
• 24.Gandikota, R., Orgad, H., Belinkov, Y., Materzyńska, J., & Bau, D. (2023). Concept Sliders: LoRA Adaptors for Precise Control in Diffusion Models. arXiv:2311.12092.
• 25.Gerstgrasser, M. et al. (2024). Is Model Collapse Inevitable? Breaking the Curse of Recursion by Accumulating Real and Synthetic Data. arXiv:2404.01413.