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+ "content": "1 INTRODUCTION"
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+ "content": ". DGMs, and most famously, diffusion models (DMs; Sohl-Dickstein et al., 2015; Ho et al., 2020), have led the \"generative AI\" boom with their ability to generate realistic images from text prompts (Karras et al., 2019; Rombach et al., 2022). DMs are thus likely to be deployed in an increasing number of public-facing or safety-critical applications. However, with sufficient model capacity, DGMs are known to memorize some of their training data. Memorization occurs at various degrees of specificity, including identities of brands, layouts of specific scenes, or exact copies of images (Webster et al., 2021; Somepalli et al., 2023a; Carlini et al., 2023)."
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+ "content": "Memorization is undesirable for myriad reasons. Simply put, the more a model reproduces its training data, the less useful it becomes. Memorization is a modelling failure under the DGM definition provided above; if the underlying ground truth "
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+ "content": " that memorizes any datapoint must be failing to generalize (Yoon et al., 2023). But memorization's risks go beyond mere utility. Training datasets may contain private information which, if memorized, might be exposed in downstream applications. Copyright law includes \"substantial similarity\" between generated and training data as a criterion in its definition of infringement, meaning that reproduced training samples can open up model builders or users to legal liability. For instance, the recent legal decision by Orrick (2023) hinged on this criterion."
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+ "content": "1. We advance the MMH in Section 2. After defining the key notions of the data manifold and local intrinsic dimension (LID), we describe how LIDs correspond to memorization."
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+ "content": "4. Finally, inspired by the MMH, in Subsection 4.2 we devise scalable approaches to avert memorization during sampling from Stable Diffusion and to identify tokens in the text conditioning that contribute to memorization."
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+ "content": "Intuition and the Manifold Hypothesis Before discussing our framework, we review some intuition relating the manifold hypothesis to practical datasets. Manifold structure "
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+ "content": ", each constraint determines a direction one cannot move without leaving the manifold and violating the structure of the dataset. Hence, a region governed by "
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+ "content": " can be intuited as the number of degrees of freedom - valid independent directions of movement in which the characteristics of the dataset are preserved. Another connection is to complexity. For example, estimates of LID from algorithms like FLIPD (Kamkari et al., 2024b) or the normal bundle (NB) method of Stanczuk et al. (2024) (which we use in our experiments; see Appendix B for details) have been shown to correspond closely with the complexity of an image; it is reasonable to expect that images with more complex features can endure more changes (such as morphing, moving, or changing the colours of different parts of the image) without losing coherence. The notions of constraints, degrees of freedom, and complexity along with their relationship to LID will help us understand its connection to memorization in later sections."
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+ "content": "A Geometric Framework for Understanding Memorization In this section we formulate a framework for understanding memorization based on comparisons between "
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+ "type": "text",
+ "content": ". As a motivating example, consider Figure 1, which depicts six possible models "
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+ "content": ". In the first scenario, Figure 1a, the"
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+ "content": "1Most authors define a manifold to have a constant dimension over the entire set. Under this common definition, our assumption is referred to as the union of manifolds hypothesis (Brown et al., 2023). We use a more general definition of manifold for brevity."
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+ "type": "text",
+ "content": " has precisely memorized some of the training data. This is a well-understood mode of memorization; training datapoints are exactly reproduced. To achieve this, the model has learned a 0-dimensional manifold around these datapoints. To our knowledge, Pidstrigach (2022) was the first to point out that a model capable of learning 0-dimensional manifolds can memorize the training data. From this example, we infer that "
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+ "type": "text",
+ "content": "However, memorization can be more complex than simply reproducing a datapoint. For example, Somepalli et al. (2023a) identify instances where layouts, styles, or foreground or background objects in training images are copied without copying the entire image, a phenomenon they refer to as reconstructive memory. Webster (2023) surfaces more instances of the same phenomenon and refers to them as template verbatim. See Figure 2 for an example. In the region of these points "
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+ "content": ", the model is able to generate images with degrees of freedom in some attributes (e.g., colour or texture), but is too constrained in other attributes (e.g., layout, style, or content). Geometrically, "
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+ "content": " is too constrained compared to the idealized ground truth manifold "
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+ "type": "text",
+ "content": ". We depict this situation in Figure 1b, wherein the model has erroneously assigned "
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+ "type": "text",
+ "content": "Two Types of Memorization We expect two types of memorization to be of interest. An academic interested in designing DGMs that learn the ground truth distribution correctly will chiefly be interested in avoiding the memorization scenario "
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+ "type": "text",
+ "content": ". We refer to this first scenario as overfitting-driven memorization (OD-Mem). This situation represents a modelling failure in that "
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+ "type": "text",
+ "content": ", and is illustrated in Figure 1a and Figure 1b."
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+ "content": "However, an industry practitioner deploying a consumer-facing model might be more interested in hypothetical values of "
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+ "content": " per se, irrespective of the values of "
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+ "content": ". For any points "
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+ "type": "text",
+ "content": " containing trademarked or private information, low values of "
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+ "content": " will be of concern even if "
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+ "content": "\\mathrm{LID}_{\\theta}(x) = \\mathrm{LID}_{*}(x)"
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+ "type": "text",
+ "content": ", as this information is likely to be revealed in samples generated from this region. A practitioner would rightly refer to this situation as memorization despite the model generalizing correctly. We refer to this second scenario as data-driven memorization (DD-Mem), and illustrate it in Figure 1d and Figure 1e. This certainly happens in practice; for example, conditioning on the title of a specific artwork (e.g. \"The Great Wave off Kanagawa\" by Katsushika Hokusai (Somepalli et al., 2023a)) is a very strong constraint, leaving few degrees of freedom in the ground truth manifold "
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+ "type": "text",
+ "content": ", but reproducing specific artworks may be undesirable in a production model. Unlike OD-Mem, DD-Mem is not overfitting in the classical sense, and a notable consequence is that it cannot be detected by comparing training and test likelihoods. We refer to the conceptualization of how LIDs relate to memorization through OD-Mem and DD-Mem as the manifold memorization hypothesis."
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+ "type": "text",
+ "content": "No memorization is present in Figure 1c, in which the model manifold "
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+ "type": "text",
+ "content": " and both have sufficiently high LID. We highlight that the MMH assumes high-performing models whose manifold "
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+ "type": "text",
+ "content": "; when this is not the case, as in Figure 1f, "
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+ "content": "3 EXPLAINING MEMORIZATION PHENOMENA"
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+ "type": "text",
+ "content": ", is common for high-dimensional data (Vahdat et al., 2021) and is necessary to generate samples from low-dimensional manifolds (Pidstrigach, 2022; Lu et al., 2023). It has been empirically observed that this explosion occurs faster as the dimensionality gap increases between the data manifold and the ambient data space (Loaiza-Ganem et al., 2024), which is one reason that generative modelling on lower-dimensional latent space tends to improve performance (Loaiza-Ganem et al., 2022). The largest "
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+ "type": "text",
+ "content": ". Hence we infer that reducing the CFG-adjusted score norm – or equivalently the CFG vector norm – should increase "
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+ "type": "text",
+ "content": " and lessen memorization, a fact confirmed empirically by Wen et al. (2023). Since this phenomenon corresponds to any "
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+ "content": " with small "
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+ "type": "text",
+ "content": ", it can indicate both OD-Mem and DD-Mem under the MMH."
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+ "content": "4 EXPERIMENTS"
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+ "content": "4.1 VERIFYING THE MANIFOLD MEMORIZATION HYPOTHESIS"
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+ "content": "In this section, we empirically verify the geometric framework which underpins the MMH. We analyze both "
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+ "type": "text",
+ "content": " for diffusion models, including the normal bundle (NB) method (Stanczuk et al., 2024), and more recently FLIPD (Kamkari et al., 2024b). For GANs, we approximate "
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+ "type": "text",
+ "content": " of generated data by computing the rank of the Jacobian of the generator. Additionally, we use LPCA (Fukunaga & Olsen, 1971) to estimate "
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+ "content": " where applicable; see Appendix B for details on these methods and Appendix C for their hyperparameter configurations. In general "
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+ "content": " are unknown quantities that are approximated with the aforementioned estimators, throughout this section we write their respective estimates as "
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+ "content": "Diffusion Model on a von Mises Mixture In an illustrative experiment, we study a mixture of a von Mises distribution, which sits on a 1-dimensional circle, and a 0-dimensional point mass at the origin in 2-dimensional ambient space, as depicted in Figure 4; every point "
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+ "content": " can be trivially replaced by either of them in the method of Wen et al. (2023). Second, we propose an automated way to use the token attributions from this method into a sample-time mitigation scheme. We start by normalizing the attributions across the tokens, and sample "
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+ "content": " tokens based on a categorical distribution parameterized by these normalized attributions. We then use GPT-4 (OpenAI, 2023) to rephrase the caption, keeping it semantically similar but perturbing the selected "
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+ "content": "Detecting and Preventing Memorization for Image Models The task of surfacing memorized samples is well-studied. Consensus in the literature is that "
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+ "content": " distance to the nearest training sample in pixel space is a poor detector of memorized samples (Carlini et al., 2023), but that recalibrating the "
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+ "content": " distance according to the local concentration of the dataset works better for smaller datasets (Yoon et al., 2023; Stein et al., 2023), and that using retrieval techniques such as distance in SSD feature space (Pizzi et al., 2022) works better still, especially for more complex, higher-resolution images (Somepalli et al., 2023a). However, all of these retrieval techniques are too expensive to be used to withhold samples from a live model. To more efficiently prevent memorized samples from being generated, past and concurrent works have altered the sampling procedure, training procedure, or the model itself (Wen et al., 2023; Daras et al., 2024; Chen et al., 2024; Hintersdorf et al., 2024)."
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+ "content": "Explaining Memorization There is an active community effort attempting to explain why and how memorization occurs in DGMs. Early studies focused on GANs, and have taken both theoretical (Nagarajan et al., 2018) and empirical (Bai et al., 2021) perspectives. However, GANs are thought to be less prone to memorization than DMs (Akbar et al., 2025), except on small datasets (Feng et al., 2021). Several works on DMs (Pidstrigach, 2022; Yi et al., 2023; Gu et al., 2023; Li et al., 2024) have pointed out that, given sufficient capacity, DMs at optimality are capable of learning the empirical training distribution, which is complete memorization. Others have focused on generalization, showing that DMs are capable of generalizing well in theory (Li et al., 2023), have inductive biases towards generating photorealistic images (Kadkhodaie et al., 2024), and will generalize when their capacity is insufficient to memorize (Yoon et al., 2023)."
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+ "content": "DGM-Based LID Estimation As opposed to statistical LID estimators (e.g., Levina & Bickel (2004)), which are constructed to estimate the dimension of "
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+ "content": ", the manifold learned by a DGM. These types of estimators are available for many types of DGMs, and in addition to being useful for memorization, have found utility in out-of-distribution detection (Kamkari et al., 2024a). In the literature, LID estimators for normalizing flows (Dinh et al., 2014) have been proposed using the singular values of their Jacobians (Horvat & Pfister, 2022; Kamkari et al., 2024a) or their density estimates (Tempczyk et al., 2022). In Section 4 we applied the singular value method to obtain LID estimates for GANs. Dai & Wipf (2019) and Zheng et al. (2022) proposed estimators for VAEs (Kingma & Welling, 2014; Rezende et al., 2014) using the structure of their posterior distribution. Several authors have proposed estimators for DMs as well (Stanczuk et al., 2024; Kamkari et al., 2024b; Horvat & Pfister, 2024); we focus on those of Stanczuk et al. (2024) and Kamkari et al. (2024b) because they work with off-the-shelf DMs and do not require modifying the training procedure."
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+ "content": "Throughout this work, we have drawn connections between the geometry of a DGM and its propensity to memorize through the MMH. First, we showed that the notion of LID provides a systematic way of understanding different types of memorization. Second, we explained how memorization phenomena described by prior work can be understood from the perspective of LID. Third, we verified the MMH empirically across scales of data and classes of models. Fourth, we showed that controlling "
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+ "type": "text",
+ "content": ". Some of these definitions define memorization globally on the level of an entire model (Meehan et al., 2020; Yoon et al., 2023; Gu et al., 2023), while others define memorization locally for individual datapoints (Carlini et al., 2023; Bhattacharjee et al., 2023). The identical definitions of Yoon et al. (2023) and Gu et al. (2023) consider a point to be memorized based purely on a distance threshold; in practice, however, distances alone have been unsuccessful at consistently surfacing what would be perceived by humans as memorized (Somepalli et al., 2023a; Stein et al., 2023). We postulate this is also due to manifold structure; semantically memorized images such as Figure 2 will sit on the same manifold, but may not necessarily be close to each other as measured by distance, even when taken in the latent space of an encoder. Meanwhile, Carlini et al. (2023) take a privacy perspective; their definition considers images memorized if they can be extracted from a model by any means, not just generated by the model. In this work we take the perspective that memorized samples are most likely to be problematic when they are generated by a production model, which are often treated as a black-box, so we focus on generation rather than extraction."
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+ "content": "Links Between Formal Memorization and the MMH For the reasons above, we use here the definition of memorization by Bhattacharjee et al. (2023), who define a point "
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+ "content": ". A natural question about the MMH is whether points satisfying it are also formally memorized by the above definition. The answer is in the negative for DD-Mem (Proposition A.2) and the affirmative for OD-Mem (Theorem A.3)."
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+ "content": " that exhibit DD-Mem at "
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+ "type": "inline_equation",
+ "content": "x_0 \\in \\mathbb{R}^d"
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+ "type": "text",
+ "content": ", but do not generate "
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+ "content": "(\\lambda, \\gamma)"
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+ "type": "text",
+ "content": "-copies of "
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+ "type": "inline_equation",
+ "content": "x_0"
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+ "type": "text",
+ "content": "."
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+ "content": " with a low-LID point "
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+ "type": "text",
+ "content": " (for example, set "
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+ "content": "\\mathrm{LID}_*(x_0) = 0"
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+ "type": "text",
+ "content": "). A perfect model will exhibit DD-Mem at "
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+ "type": "text",
+ "content": ", but for any ball "
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+ "type": "inline_equation",
+ "content": "B_r^d(x_0)"
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+ {
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+ "type": "text",
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+ "content": "P_{\\theta}(B_r^d(x_0)) = P_{*}(B_r^d(x_0))"
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+ "type": "text",
+ "content": ", violating the second condition of Definition A.1."
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+ "content": "-copies of "
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+ "content": "x_0"
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+ "type": "text",
+ "content": " with near-certainty."
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+ "content": "-copies. The first is that Definition A.1 is in some sense incomplete; it does not cover DD-Mem. The second is that OD-Mem can be considered a useful refinement of Definition A.1. While the algorithm given by Bhattacharjee et al. (2023) is intractable at scale, we show in Section 4 that the added strength (in the mathematical sense) of the MMH allows us to flag memorized data more efficiently using only estimators of LID."
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+ "type": "text",
+ "content": ", by approximating it with a neural network "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "s_\\theta : \\mathbb{R}^d \\times (0,1] \\to \\mathbb{R}^d"
+ },
+ {
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+ "type": "text",
+ "content": ". Once the network is trained, "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "s_\\theta(x_t,t) \\approx \\nabla \\log p_t(x_t)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " is plugged in Equation 4, and solving the resulting SDE transforms noise into model samples. Below we briefly summarize two existing methods, FLIPD and NB, for approximating "
+ },
+ {
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+ "type": "inline_equation",
+ "content": "\\mathrm{LID}_\\theta(x)"
+ },
+ {
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+ "type": "text",
+ "content": " for a DM."
+ }
+ ]
+ }
+ ],
+ "index": 7
+ },
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+ "type": "text",
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+ "spans": [
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+ "bbox": [
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+ "type": "text",
+ "content": "FLIPD Kamkari et al. (2024b) proposed FLIPD, an estimator of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathrm{LID}_{\\theta}(x)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " for DMs. Commonly "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "f"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " is linear in "
+ },
+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "x_{t}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", in which case the transition kernel corresponding to the forward SDE is given by"
+ }
+ ]
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
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+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
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+ "spans": [
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+ "type": "interline_equation",
+ "content": "p _ {t \\mid 0} \\left(x _ {t} \\mid x _ {0}\\right) = \\mathcal {N} \\left(x _ {t}; \\psi (t) x _ {0}, \\sigma^ {2} (t) I _ {d}\\right), \\tag {5}",
+ "image_path": "a5aba19178868ea84b9946964be08a462d08ffb8f5850ce1c75cd87adafe6672.jpg"
+ }
+ ]
+ }
+ ],
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+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
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+ ],
+ "spans": [
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "where "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\psi, \\sigma: [0,1] \\to \\mathbb{R}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " are known functions which depend on the choices of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "f"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " and "
+ },
+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "g"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", and which can be easily evaluated. For a DM with such a transition kernel, FLIPD is defined as"
+ }
+ ]
+ }
+ ],
+ "index": 10
+ },
+ {
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+ "type": "interline_equation",
+ "angle": 0,
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+ "spans": [
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+ "type": "interline_equation",
+ "content": "\\operatorname {F L I P D} (x, t _ {0}) = d + \\sigma^ {2} (t _ {0}) \\left(\\operatorname {t r} \\left(\\nabla s _ {\\theta} (\\psi (t _ {0}) x, t _ {0})\\right) + \\| s _ {\\theta} (\\psi (t _ {0}) x, t _ {0}) \\| ^ {2}\\right), \\tag {6}",
+ "image_path": "c23be7f55fb70fd22c54257e4c38d06448b412d352b511869c9203247937ce50.jpg"
+ }
+ ]
+ }
+ ],
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+ },
+ {
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+ "type": "text",
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+ "type": "text",
+ "content": "where "
+ },
+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "t_0 \\in [0,1]"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " is a hyperparameter. Kamkari et al. (2024b) proved that, when "
+ },
+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "t_0 \\approx 0"
+ },
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+ "type": "text",
+ "content": " and "
+ },
+ {
+ "bbox": [
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+ "type": "inline_equation",
+ "content": "x \\in \\mathcal{M}_{\\theta}"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": ", "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathrm{FLIPD}(x,t_0)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " is a valid approximation of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathrm{LID}_{\\theta}(x)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". The reason for this is that the rate of change of the log density of the convolution between "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "p_{\\theta}(x)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " and a Gaussian evaluated at "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "x_0"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " with respect to the amount of added Gaussian noise approximates "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathrm{LID}_{\\theta}(x_0)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", and Kamkari et al. (2024b) showed that FLIPD computes this rate of change. In practice, computing the trace of the Jacobian of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "s_{\\theta}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " is the only expensive operation needed to compute FLIPD, and this is easily approximated by using the Hutchinson stochastic trace estimator (Hutchinson, 1989)."
+ }
+ ]
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
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+ "type": "text",
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+ ],
+ "spans": [
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+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "NB Stanczuk et al. (2024) proposed another estimator of "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathrm{LID}_{\\theta}(x)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " for DMs. Following Kamkari et al. (2024b), we refer to this estimator as the normal bundle (NB) estimator. Stanczuk et al. (2024) proved that when "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "f(x_{t},t)\\equiv 0"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "s_\\theta (x_t,t)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " points orthogonally towards "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{M}_{\\theta}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " as "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "t\\to 0"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". They leverage this observation as follows: for a given "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "x"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", Equation 3 is started at "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "x"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " and run forward until time "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "t_0"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "; this is done "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "k"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " times, resulting in "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "x_{t_0}^{(1)},\\ldots ,x_{t_0}^{(k)}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". The matrix "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "S_{\\theta}(x,t_0)\\in \\mathbb{R}^{d\\times k}"
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "content": " is then constructed as"
+ }
+ ]
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
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+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
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+ "spans": [
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+ "type": "interline_equation",
+ "content": "S _ {\\theta} (x, t _ {0}) = \\left[ s _ {\\theta} \\left(x _ {t _ {0}} ^ {(1)}, t _ {0}\\right) \\mid \\dots \\mid s _ {\\theta} \\left(x _ {t _ {0}} ^ {(k)}, t _ {0}\\right) \\right], \\tag {7}",
+ "image_path": "09c1198eea92f72a1d14fc6dff097c9ca924cb063697bcb418e6c97d8a5538f7.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 14
+ },
+ {
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+ "type": "text",
+ "angle": 0,
+ "lines": [
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+ "spans": [
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+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "and thanks to the previous observation, the columns of "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "S_{\\theta}(x,t_0)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " approximately span the normal space of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{M}_{\\theta}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " at "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "x"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " when "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "t_0 \\approx 0"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", meaning that rank "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "S_{\\theta}(x,t_0) \\approx d - \\mathrm{LID}_{\\theta}(x)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". The NB estimator is given by"
+ }
+ ]
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
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+ ],
+ "spans": [
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+ ],
+ "type": "interline_equation",
+ "content": "\\mathrm {N B} (x, t _ {0}) = d - \\operatorname {r a n k} S _ {\\theta} (x, t _ {0}). \\tag {8}",
+ "image_path": "c163da3fc444d8ae948c99107cbc6aaafbe24eaecbb93e9966cb58ac483d3975.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 16
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
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+ ],
+ "spans": [
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "In practice the rank is numerically computed by setting a threshold, carrying out a singular value decomposition of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "S_{\\theta}(x,t_0)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", and counting the number of singular values above the threshold. Stanczuk et al. (2024) recommend setting "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "k = 4d"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", and we follow this recommendation. Computing the NB estimator is much more expensive than FLIPD, since "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "4d"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " forward calls have to be made to construct "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "S_{\\theta}(x,t_0)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", and then the singular value decomposition has a cost which is cubic in "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "d"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". Finally, we point out that when "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "f"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " is not identically equal to 0, the NB method can be easily adapted to still provide a valid approximation of "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathrm{LID}_{\\theta}(x)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " (Kamkari et al., 2024a)."
+ }
+ ]
+ }
+ ],
+ "index": 17
+ },
+ {
+ "bbox": [
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+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
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+ ],
+ "spans": [
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "We highlight that both FLIPD and NB were originally developed as estimators of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathrm{LID}_*(x)"
+ },
+ {
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+ "content": " in decoder-based DGMs (Horvat & Pfister, 2022; Kamkari et al., 2024a; Humayun et al., 2024)."
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+ "content": "Figure 8: Removing and analyzing image complexity as a confounding factor in memorization detection for CIFAR10 (a-b) and Stable Diffusion (c)."
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+ "type": "text",
+ "content": " with Local PCA As established in Appendix B.2, local PCA estimates the intrinsic dimensionality of a datapoint by counting the number of significant explained variances from PCA performed on the datapoint's local neighbourhood, determined by its "
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+ "type": "text",
+ "content": " in our experiments). Finding the significant explained variances is done through a threshold hyperparameter, "
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+ "type": "text",
+ "content": " are considered significant. For our approach in Figure 5d, we introduce two key modifications to better adapt the original Local PCA algorithm for detecting DD-Mem: "
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+ "content": " instead of selecting "
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+ "content": " individually for each datapoint, we define it globally as the 10th percentile of all explained variances across the entire dataset; "
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+ "content": " if a datapoint has neighbours within the 10th percentile of all pairwise distances, we restrict the neighbourhood to those points. The second modification allows us to avoid including distant points in the neighbourhood if closer ones already exist and especially helps us detect zero-dimensional point masses."
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+ "content": " as the 10th percentile of all singular values computed from the generated images in Figure 5b and Figure 8b."
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+ "content": " for FLIPD and use the Hutchinson trace estimator to approximate the trace of the score gradient in Equation 6. In line with Kamkari et al. (2024b), we apply this in the latent space of Stable diffusion and use a single Hutchinson sample to estimate Equation 6 for all of our large-scale experiments."
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+ "content": "CFG Norm for Detecting Memorized Samples in the Training Set Note that while Wen et al. (2023) use the generation process to measure whether a synthesized image has been memorized, we were interested in detecting whether real, training-set images have been memorized in Figure 6, which requires some methodological changes. To compute a memorization score, we take "
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+ "type": "text",
+ "content": " with the probability flow ODE (Song et al., 2021) until time "
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+ "content": "\\| s_{\\theta}(x_0; t_0, c) - s_{\\theta}(x; t_0, \\emptyset) \\|"
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+ "content": " using FLIPD with DDPMs (recall Equation 6)."
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+ "content": "When sampling from DMs we use the DDIM sampler (Song et al., 2020), mirroring the setup in Wen et al. (2023). In our notation, this sampler defines "
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+ "content": " is given as in Equation 3. In turn, "
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+ "content": " obeys the forward SDE:"
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+ "content": "\\mathrm {d} \\tilde {x} _ {t} = \\tilde {g} (t) \\mathrm {d} W _ {t}, \\quad \\tilde {x} _ {0} \\sim p _ {*} (x), \\tag {11}",
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+ "content": "\\tilde {\\epsilon} _ {\\theta} (x, t) = - \\tilde {\\sigma} (t / T) \\tilde {s} _ {\\theta} (x, t / T). \\tag {13}",
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+ "content": "We highlight that FLIPD (Equation 6) can be applied using the forward SDE in Equation 11 along with its corresponding score function in Equation 12, resulting in the estimate"
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+ "content": "\\widetilde {\\operatorname {F L I P D}} (x, t) = d + \\tilde {\\sigma} ^ {2} (t) \\left(\\operatorname {t r} \\left(\\nabla \\tilde {s} _ {\\theta} (x, t)\\right) + \\| \\tilde {s} _ {\\theta} (x, t) \\| ^ {2}\\right). \\tag {14}",
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+ "content": " thanks to Equation 13. We also note that in our text-conditioning analysis, we are interested in the probabilities conditioned by the text prompt, thus, these score functions are extended by the conditioning variable "
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+ "type": "text",
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+ "content": "D.2 UNIFYING DIFFERENTIABLE METRICS FOR TEXT-CONDITIONED MEMORIZATION"
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+ "content": "We begin by revisiting the differentiable memorization metric used by Wen et al. (2023) for detecting and mitigating memorization, reformulating it within the continuous, score-based framework of diffusion models. Building on this, we perform an analysis, making minimal modifications to the original formulation to derive alternative metrics that remain effective and are theoretically-grounded. As a result, here we will formally derive three differentiable metrics: "
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+ "content": ". We show that the value Wen et al. (2023) compute in their paper is in fact an estimator of "
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+ "content": ", rescaled by a constant. We then make minor modifications to introduce the two new metrics "
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+ "content": " and interpret them through the lens of the MMH."
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+ "content": "The Differentiable Metric of Wen et al. (2023) For any text condition "
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+ "content": "th sample following the (DDIM) trajectory "
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+ "content": "\\{\\tilde{x}_T^{(n)}, \\tilde{x}_{T-1}^{(n)}, \\ldots, \\tilde{x}_0^{(n)}\\}"
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+ "type": "text",
+ "content": " from noise to data through the denoising process. They then introduce the following metric, which we have slightly reformulated to match our notation:"
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+ "content": "\\mathcal {A} ^ {\\mathrm {C F G}} (c; N, T) = \\frac {1}{T N} \\sum_ {n = 1} ^ {N} \\sum_ {t = 0} ^ {T} \\left\\| \\tilde {\\epsilon} _ {\\theta} \\left(\\tilde {x} _ {t} ^ {(n)}; t, c\\right) - \\tilde {\\epsilon} _ {\\theta} \\left(\\tilde {x} _ {t} ^ {(n)}; t, \\emptyset\\right) \\right\\| ^ {2}. \\tag {15}",
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+ "content": "We colour-code the metric in red to distinguish between it and the analogous metric that we will shortly derive at the end of this section."
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+ "content": " represent the marginal probability at time "
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+ "content": " induced by the DDIM sampler conditioned on "
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+ "type": "text",
+ "content": " with the addition of the CFG term. Recall from Equation 2 that the score used for sampling from "
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+ "content": "\\tilde {s} _ {\\theta} ^ {\\mathrm {C F G}} (x; t, c) = \\tilde {s} _ {\\theta} (x; t, \\emptyset) + \\lambda (\\tilde {s} _ {\\theta} (x; t, c) - \\tilde {s} _ {\\theta} (x; t, \\emptyset)). \\tag {16}",
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+ "content": "\\mathcal {A} ^ {\\mathrm {C F G}} (c; N, T) := \\frac {1}{T N} \\sum_ {n = 1} ^ {N} \\sum_ {t = 0} ^ {T} \\left\\| - \\frac {\\tilde {\\sigma} (t / T)}{\\lambda} \\left[ \\tilde {s} _ {\\theta} ^ {\\mathrm {C F G}} \\left(\\tilde {x} _ {t} ^ {(n)}; t / T, c\\right) - \\tilde {s} _ {\\theta} \\left(\\tilde {x} _ {t} ^ {(n)}; t / T, \\emptyset\\right) \\right] \\right\\| ^ {2}. \\tag {17}",
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+ "content": " denotes the uniform distribution. Next, we observe that the inner term of the expectation on the right-hand-side of Equation 21 is in fact a Monte-Carlo estimator. By the law of large numbers, we have the following:"
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+ "content": "We now see that with the new formulation, all the red terms in Equation 23, have gone away, making it fully amenable to the score-based formulation of diffusion models. The "
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+ "type": "text",
+ "content": " factor merely scales the metric, and for the purposes of detection and mitigation, this scaling is inconsequential: if a metric effectively predicts memorization, rescaling it will not diminish its effectiveness as a predictor. We thus disregard the scaling factor "
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+ "content": " of timesteps in "
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+ "content": "; this would allow our metric to be a generalization of the one proposed by Wen et al. (2023):"
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+ "content": "Simplifying Further We have shown that the CFG vector norm and the CFG adjusted score norm behave similarly in Figure 3. If, instead of considering the CFG vector norm in Equation 24, we consider the CFG-adjusted score "
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+ "content": "We have shown this to be a viable memorization metric, able to detect tokens driving memorization in Figure 11, and behaving comparable to "
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+ "content": ", tends to become large, causing the metric in Equation 25 to increase significantly."
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+ "content": ". Recalling Equation 6, we can define the class-conditional "
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+ "content": " estimate based on FLIPD as follows, analogously to Equation 14:"
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+ "content": "\\widetilde {\\text {F L I P D}} ^ {\\text {C F G}} (x; t, c) = d + \\tilde {\\sigma} ^ {2} (t) \\cdot \\left(\\operatorname {t r} \\left(\\nabla \\tilde {s} _ {\\theta} ^ {\\text {C F G}} (x; t, c)\\right) + \\| \\tilde {s} _ {\\theta} ^ {\\text {C F G}} (x; t, c) \\| ^ {2}\\right). \\tag {26}",
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+ "content": "Noting that "
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+ "type": "text",
+ "content": " has a similar term to Equation 25, we add "
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+ "type": "text",
+ "content": " and the trace term from Equation 26 into Equation 25, and propose the following MMH-based metric:"
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+ "content": "d + \\mathcal {A} ^ {s _ {\\theta} ^ {\\mathrm {C F G}}} (c) + \\mathbb {E} _ {t \\sim \\mathcal {T} (0, 1)} \\mathbb {E} _ {\\tilde {x} _ {t} \\sim \\tilde {p} _ {t} ^ {\\mathrm {C F G}} (\\cdot | c)} \\left[ \\tilde {\\sigma} ^ {2} (t) \\cdot \\operatorname {t r} \\left(\\nabla \\tilde {s} _ {\\theta} ^ {\\mathrm {C F G}} \\left(\\tilde {x} _ {t}; t, c\\right)\\right) \\right] = \\tag {27}",
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+ "type": "text",
+ "content": " are "
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+ "type": "text",
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+ "type": "inline_equation",
+ "content": "\\mathcal{M}_\\theta \\subset \\mathbb{R}^d"
+ },
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+ "bbox": [
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+ "type": "text",
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+ "type": "inline_equation",
+ "content": "\\mathbb{R}^d"
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+ "type": "text",
+ "content": " with metrics "
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+ "content": "g_*"
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+ "type": "text",
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+ "content": "g_\\theta"
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+ "type": "text",
+ "content": " respectively. We denote the Riemannian measures on "
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+ "content": "\\mathcal{M}_*"
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+ "type": "text",
+ "content": " and "
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+ "content": "\\mathcal{M}_\\theta"
+ },
+ {
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+ "type": "text",
+ "content": " as "
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+ "content": "\\mu_*"
+ },
+ {
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+ "type": "text",
+ "content": " and "
+ },
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+ ],
+ "type": "inline_equation",
+ "content": "\\mu_\\theta"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", respectively, so that "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "p_*(x) = \\mathrm{d}P_* / \\mathrm{d}\\mu_*(x)"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": " and "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "p_\\theta(x) = \\mathrm{d}P_\\theta / \\mathrm{d}\\mu_\\theta(x)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". As mentioned in Section 2, we take a lax definition of manifold which allows them to vary in dimensionality in different components. A single manifold under our definition is equivalent to a disjoint union of manifolds under the more standard definition."
+ }
+ ]
+ }
+ ],
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+ },
+ {
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+ "type": "title",
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+ "type": "text",
+ "content": "E.1 PROPOSITION 3.1"
+ }
+ ]
+ }
+ ],
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+ },
+ {
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+ "type": "text",
+ "content": "Lemma E.1. Assume that "
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+ "content": "p_{*}(x) > 0"
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+ "type": "text",
+ "content": " for every "
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+ "type": "text",
+ "content": ", and let "
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+ "type": "text",
+ "content": ". Then, the following are equivalent:"
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+ ],
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+ {
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+ "type": "text",
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+ "type": "text",
+ "content": "Proof."
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+ "content": "(1) "
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+ "type": "text",
+ "content": "."
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+ "type": "interline_equation",
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+ "type": "text",
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+ "type": "text",
+ "content": ". If we had "
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+ "content": "\\mathrm{LID}_{*}(x_0) > 0"
+ },
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+ "type": "text",
+ "content": " this would incur a contradiction: letting "
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+ "type": "inline_equation",
+ "content": "(U,\\phi)"
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+ "type": "text",
+ "content": " be a chart around "
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+ "type": "text",
+ "content": ", then by the definition of "
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+ "type": "interline_equation",
+ "content": "0 < \\mu_ {*} (\\{x _ {0} \\}) = \\int_ {\\phi (\\{x _ {0} \\})} \\sqrt {\\det (g _ {*})} d \\lambda , \\tag {30}",
+ "image_path": "f71bf5bb0966bf1a799bfd910c99a17cec132ab671c692f98f9945d8c31847b2.jpg"
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+ }
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+ "type": "text",
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+ "type": "inline_equation",
+ "content": "\\lambda"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": " is the Lebesgue measure on "
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+ "type": "inline_equation",
+ "content": "\\mathbb{R}^{\\mathrm{LID}_*(x_0)}"
+ },
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+ "type": "text",
+ "content": " or the counting measure if "
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+ "type": "text",
+ "content": ". Due to the singleton domain of integration, positivity of the integral in Equation 30 would be impossible unless "
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+ {
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+ "type": "text",
+ "content": "."
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+ ],
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+ "type": "text",
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+ "content": "\\Rightarrow"
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+ "type": "text",
+ "content": " (1) Suppose "
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+ "content": "\\mathrm{LID}(x_0) = 0"
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+ "type": "text",
+ "content": ". This implies that "
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+ "content": "\\{x_0\\}"
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+ "type": "text",
+ "content": " is an open set in the subspace topology of "
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+ "content": "\\mathcal{M}_*"
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+ "type": "text",
+ "content": ". Since "
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+ "type": "inline_equation",
+ "content": "x_0 \\in \\operatorname{supp} \\mu_*"
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+ "type": "text",
+ "content": ", any open set containing "
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+ "type": "inline_equation",
+ "content": "x_0"
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+ "type": "text",
+ "content": " must have positive measure under "
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+ "type": "text",
+ "content": ", so that "
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+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\mu_*(\\{x_0\\}) > 0"
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+ {
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+ ],
+ "type": "text",
+ "content": ". Then, since "
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+ {
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+ ],
+ "type": "inline_equation",
+ "content": "P_*(\\{x_0\\}) = p_*(x_0) \\mu_*(\\{x_0\\})"
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+ {
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+ "type": "text",
+ "content": " and "
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+ "type": "text",
+ "content": ", it follows that "
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+ "type": "text",
+ "content": "."
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+ }
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+ }
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+ "type": "text",
+ "content": "Proposition E.2 (Formal Restatement of Proposition 3.1). Assume that "
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+ "content": "p_{*}(x) > 0"
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+ "content": " for every "
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+ "content": "x \\in \\mathcal{M}_{*}"
+ },
+ {
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+ ],
+ "type": "text",
+ "content": ". Let "
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+ {
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+ ],
+ "type": "inline_equation",
+ "content": "\\{x_{i}\\}_{i=1}^{n}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " be a training dataset drawn independently from "
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+ "type": "inline_equation",
+ "content": "p_{*}(x)"
+ },
+ {
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+ "type": "text",
+ "content": ". Then:"
+ }
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+ {
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+ "type": "text",
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+ {
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+ "type": "text",
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+ "content": "L I D _ {\\theta} \\left(x _ {0} \\mid c\\right) \\leq L I D _ {\\theta} \\left(x _ {0}\\right). \\tag {38}",
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+ "type": "text",
+ "content": "Proof. If "
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+ "content": "S(c)"
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+ "type": "text",
+ "content": " is a submanifold of "
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+ "type": "inline_equation",
+ "content": "\\mathbb{R}^d"
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+ {
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+ "type": "text",
+ "content": ", then it is also a submanifold of "
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+ "type": "inline_equation",
+ "content": "\\mathcal{M}_{\\theta}"
+ },
+ {
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+ "type": "text",
+ "content": ". The inequality follows directly."
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+ "type": "text",
+ "content": "E.3 THEOREM A.3"
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+ }
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+ "type": "text",
+ "content": "Here we show that OD-mem implies data-copying under the definition of Bhattacharjee et al. (2023)."
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+ "type": "inline_equation",
+ "content": "d_0"
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+ {
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+ "type": "text",
+ "content": "-dimensional smooth Riemannian submanifold of Euclidean space "
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+ {
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+ "type": "inline_equation",
+ "content": "\\mathbb{R}^d"
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+ {
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+ "type": "text",
+ "content": ". Let "
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+ {
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+ "type": "text",
+ "content": " be the Riemannian measure of "
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+ "content": "\\mathcal{M}"
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+ "type": "text",
+ "content": ". If "
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+ "content": "B_r^d (x_0)"
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+ "content": " denotes the "
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+ {
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+ "type": "text",
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+ "type": "text",
+ "content": ", then there exist constants "
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+ "content": " and "
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+ "content": " not depending on "
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+ "content": " such that for all small enough "
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+ "type": "text",
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+ "content": "Proof. Without loss of generality, by rotating and translating, we assume "
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+ "type": "text",
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+ ],
+ "type": "text",
+ "content": "be the graph of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "u"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " in the closed "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "d_0"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "-ball "
+ },
+ {
+ "bbox": [
+ 104,
+ 120,
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+ 144
+ ],
+ "type": "inline_equation",
+ "content": "B_r^{d_0}(0)"
+ },
+ {
+ "bbox": [
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+ 144
+ ],
+ "type": "text",
+ "content": ". Note that both are defined when "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "u"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " is defined, i.e. small enough "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "r"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", and are subset of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\mathcal{M}"
+ },
+ {
+ "bbox": [
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+ 120,
+ 506,
+ 144
+ ],
+ "type": "text",
+ "content": ". Then it is clear that we have"
+ }
+ ]
+ }
+ ],
+ "index": 3
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
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+ ],
+ "spans": [
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+ "bbox": [
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+ ],
+ "type": "interline_equation",
+ "content": "B _ {r} ^ {d} \\left(x _ {0}\\right) \\cap \\mathcal {M} \\subseteq G (r). \\tag {43}",
+ "image_path": "c8a722ed222b7e31b65452325436f6df10df8fb1d5e5468fbf2f8a546d7e18f6.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 4
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "angle": 0,
+ "lines": [
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+ "bbox": [
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+ ],
+ "spans": [
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+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "Now, consider the function "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "v(z) = \\frac{\\|u(z)\\|}{\\|z\\|}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". Note that "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "v(z)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " is continuous everywhere in "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "B_r^{d_0}(0) \\setminus \\{0\\}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". Since "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "u"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " and its derivatives vanish at "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "z = 0"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", from the definition, we have "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\lim_{z \\to 0} v(z) = 0"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " as well. Thus "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "v(z)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " can be extended to a continuous function in "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "B_r^{d_0}(0)"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ". Fix "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "R > 0"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", and let "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "K"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " be the maximum of "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "v"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " over "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\overline{B_R^{d_0}}"
+ },
+ {
+ "bbox": [
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+ 224
+ ],
+ "type": "text",
+ "content": ". Then, if "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "\\|z\\| < ar"
+ },
+ {
+ "bbox": [
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+ 224
+ ],
+ "type": "text",
+ "content": " where "
+ },
+ {
+ "bbox": [
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+ 224
+ ],
+ "type": "inline_equation",
+ "content": "a = \\frac{1}{\\sqrt{1 + K^2}}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": ", we have"
+ }
+ ]
+ }
+ ],
+ "index": 5
+ },
+ {
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+ ],
+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
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+ ],
+ "spans": [
+ {
+ "bbox": [
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+ ],
+ "type": "interline_equation",
+ "content": "\\left\\| z \\right\\| ^ {2} + \\left\\| u (z) \\right\\| ^ {2} = \\left\\| z \\right\\| ^ {2} \\left(1 + v (z) ^ {2}\\right) \\leq \\left\\| z \\right\\| ^ {2} \\left(1 + K ^ {2}\\right) < r ^ {2}. \\tag {44}",
+ "image_path": "ae1a67df2c80eda733185a94d31f13a8f7f9b1a317851502ab56734e972a4aaa.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 6
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
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+ ],
+ "spans": [
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+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "This shows that "
+ },
+ {
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+ ],
+ "type": "inline_equation",
+ "content": "G(ar) \\subseteq B_r^d(x_0) \\cap \\mathcal{M}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " for "
+ },
+ {
+ "bbox": [
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+ 258
+ ],
+ "type": "inline_equation",
+ "content": "0 < r < R"
+ },
+ {
+ "bbox": [
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+ 245,
+ 389,
+ 258
+ ],
+ "type": "text",
+ "content": ". Thus we have"
+ }
+ ]
+ }
+ ],
+ "index": 7
+ },
+ {
+ "bbox": [
+ 217,
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+ ],
+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
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+ ],
+ "spans": [
+ {
+ "bbox": [
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+ ],
+ "type": "interline_equation",
+ "content": "\\mu (G (a r)) \\leq \\mu \\left(B _ {r} ^ {d} \\left(x _ {0}\\right) \\cap \\mathcal {M}\\right) \\leq \\mu (G (r)). \\tag {45}",
+ "image_path": "892af603b6305412d9627a500c5acf47e3637ac7f299f384cdce16f1cb28afab.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 8
+ },
+ {
+ "bbox": [
+ 104,
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+ ],
+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
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+ ],
+ "spans": [
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": "Let "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "K_{1}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " and "
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "inline_equation",
+ "content": "K_{2}"
+ },
+ {
+ "bbox": [
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+ ],
+ "type": "text",
+ "content": " be the minimum and maximum of "
+ },
+ {
+ "bbox": [
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+ 504,
+ 297
+ ],
+ "type": "inline_equation",
+ "content": "\\sqrt{\\operatorname*{det}g}"
+ },
+ {
+ "bbox": [
+ 104,
+ 283,
+ 504,
+ 297
+ ],
+ "type": "text",
+ "content": " over "
+ },
+ {
+ "bbox": [
+ 104,
+ 283,
+ 504,
+ 297
+ ],
+ "type": "inline_equation",
+ "content": "B_r^{d_0}(0)"
+ },
+ {
+ "bbox": [
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+ 283,
+ 504,
+ 297
+ ],
+ "type": "text",
+ "content": ", respectively. Then we have"
+ }
+ ]
+ }
+ ],
+ "index": 9
+ },
+ {
+ "bbox": [
+ 180,
+ 300,
+ 505,
+ 327
+ ],
+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
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+ 300,
+ 505,
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+ ],
+ "spans": [
+ {
+ "bbox": [
+ 180,
+ 300,
+ 505,
+ 327
+ ],
+ "type": "interline_equation",
+ "content": "\\mu (G (r)) = \\int_ {B _ {r} ^ {d _ {0}} (0)} \\sqrt {\\det g} \\mathrm {d} z \\leq \\int_ {B _ {r} ^ {d _ {0}} (0)} K _ {2} \\mathrm {d} z = K _ {2} V _ {d _ {0}} r ^ {d _ {0}} \\tag {46}",
+ "image_path": "bf8cc9e3782678142c5fb5447db18529ad6c335f89eacd4238d2a94ee5b4b43f.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 10
+ },
+ {
+ "bbox": [
+ 105,
+ 331,
+ 124,
+ 341
+ ],
+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
+ 105,
+ 331,
+ 124,
+ 341
+ ],
+ "spans": [
+ {
+ "bbox": [
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+ 124,
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+ ],
+ "type": "text",
+ "content": "and"
+ }
+ ]
+ }
+ ],
+ "index": 11
+ },
+ {
+ "bbox": [
+ 179,
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+ 365
+ ],
+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
+ 179,
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+ 505,
+ 365
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 179,
+ 339,
+ 505,
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+ ],
+ "type": "interline_equation",
+ "content": "\\mu (G (r)) = \\int_ {B _ {r} ^ {d _ {0}} (0)} \\sqrt {\\det g} d z \\geq \\int_ {B _ {r} ^ {d _ {0}} (0)} K _ {1} d z = K _ {1} V _ {d _ {0}} r ^ {d _ {0}}, \\tag {47}",
+ "image_path": "a86055b7db9730dd2a39b1cb46e65cf629a8942d65f0922d103a54751c8c377d.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 12
+ },
+ {
+ "bbox": [
+ 104,
+ 367,
+ 479,
+ 380
+ ],
+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 367,
+ 479,
+ 380
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 367,
+ 479,
+ 380
+ ],
+ "type": "text",
+ "content": "where "
+ },
+ {
+ "bbox": [
+ 104,
+ 367,
+ 479,
+ 380
+ ],
+ "type": "inline_equation",
+ "content": "V_{d_0}"
+ },
+ {
+ "bbox": [
+ 104,
+ 367,
+ 479,
+ 380
+ ],
+ "type": "text",
+ "content": " is the Euclidean volume of the unit "
+ },
+ {
+ "bbox": [
+ 104,
+ 367,
+ 479,
+ 380
+ ],
+ "type": "inline_equation",
+ "content": "d_0"
+ },
+ {
+ "bbox": [
+ 104,
+ 367,
+ 479,
+ 380
+ ],
+ "type": "text",
+ "content": "-ball. Combining the above results, we have"
+ }
+ ]
+ }
+ ],
+ "index": 13
+ },
+ {
+ "bbox": [
+ 157,
+ 384,
+ 505,
+ 398
+ ],
+ "type": "interline_equation",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
+ 157,
+ 384,
+ 505,
+ 398
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 157,
+ 384,
+ 505,
+ 398
+ ],
+ "type": "interline_equation",
+ "content": "K _ {1} V _ {d _ {0}} a ^ {d _ {0}} r ^ {d _ {0}} \\leq \\mu (G (a r)) \\leq \\mu \\left(B _ {r} ^ {d} \\left(x _ {0}\\right) \\cap \\mathcal {M}\\right) \\leq \\mu (G (r)) \\leq K _ {2} V _ {d _ {0}} r ^ {d _ {0}}, \\tag {48}",
+ "image_path": "067010f4b042ef888e483f34c1128cb0e769c6540fe836ae6e980e0a80441ade.jpg"
+ }
+ ]
+ }
+ ],
+ "index": 14
+ },
+ {
+ "bbox": [
+ 104,
+ 401,
+ 379,
+ 415
+ ],
+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 401,
+ 379,
+ 415
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 401,
+ 379,
+ 415
+ ],
+ "type": "text",
+ "content": "which finishes the proof with "
+ },
+ {
+ "bbox": [
+ 104,
+ 401,
+ 379,
+ 415
+ ],
+ "type": "inline_equation",
+ "content": "C_1^{\\mathcal{M}} = K_1V_{d_0}a^{d_0}"
+ },
+ {
+ "bbox": [
+ 104,
+ 401,
+ 379,
+ 415
+ ],
+ "type": "text",
+ "content": " and "
+ },
+ {
+ "bbox": [
+ 104,
+ 401,
+ 379,
+ 415
+ ],
+ "type": "inline_equation",
+ "content": "C_2^{\\mathcal{M}} = K_2V_{d_0}"
+ },
+ {
+ "bbox": [
+ 104,
+ 401,
+ 379,
+ 415
+ ],
+ "type": "text",
+ "content": "."
+ }
+ ]
+ }
+ ],
+ "index": 15
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "angle": 0,
+ "lines": [
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "spans": [
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": "Theorem E.5 (Formal Restatement of Theorem A.3). Assume that "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "inline_equation",
+ "content": "p_{*}(x)"
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": " and "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "inline_equation",
+ "content": "p_{\\theta}(x)"
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": " are continuous and that "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "inline_equation",
+ "content": "p_{\\theta}(x)"
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": " is strictly positive. Let "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "inline_equation",
+ "content": "x_0 \\in \\mathcal{M}_{\\theta} \\cap \\mathcal{M}_*"
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": " and let "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "inline_equation",
+ "content": "p_{\\theta}(x)"
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": " be a model undergoing OD-mem at "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "inline_equation",
+ "content": "x_0"
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": ", i.e. "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "inline_equation",
+ "content": "0 \\leq LID_{\\theta}(x_0) < LID_{*}(x_0)"
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": ". Then for any "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "inline_equation",
+ "content": "\\lambda > 1"
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": " and "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "inline_equation",
+ "content": "0 < \\gamma < 1"
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
+ "type": "text",
+ "content": ", there exists a radius "
+ },
+ {
+ "bbox": [
+ 104,
+ 423,
+ 504,
+ 491
+ ],
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+ "content": "r_0"
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