In the main loop (lines 5-9), the D terms in F(\x) are computed one-by-one. However, most of the lost information should not be noticeable to visitors when using a balanced compression ratio. For example, VAEs suffer from a compression cost overhead due to their latent variables. 0 share Despite extensive progress on image generation, deep generative models are suboptimal when applied to lossless compression. EiNets and RAT-SPNs are also balanced since they also have an equivalent PGM representation of their PCs. Our results highlight the potential impact that non-standard learning architectures may have on neural data compression. In this paper, we propose a learned lossless compression system by leveraging the power of the lossy BPG, as illustrated in Fig. Thus, we have prove that Eq. 0 forks Releases No releases published. However, due to the irregular structure of learned CLTs, HCLTs cannot be easily vectorized. Parameter learning was performed by the following steps. Mondays 16:15-17:45 and Tuesdays 12:15-13:45 on zoom. This is done by (i) assigning probabilities to the input units \wrtthe given evidence x1, x2, and x4 (assign 0 to the input unit labeled X2 and X4 as they contradict the given evidence; all other input units are assigned probability 1), and (ii) evaluate the probabilities of sum/product units following Eq. PCs guarantee tractable computation of a query class, e.g., marginals, MAP inference, expectations, etc., if their computational graphs satisfy certain well-defined properties. Before proceeding with a formal argument, we give a high-level explanation of the acceleration. However, PP can only prune away units in Group #2. Donate or volunteer today! 7(a)) is converted into an ordered PC (Fig. 1.Specifically, we decompose the input image x into the lossy reconstruction x l produced by BPG and the corresponding residual r.We then learn a probabilistic model p (r | x l) of the residual, conditionally on the lossy reconstruction x l. Consider using order =(X3,X2,X1), as shown in Fig. 4 watching Forks. Van Schaik (2017), EMNIST: extending mnist to handwritten letters, 2017 International Joint Conference on Neural Networks (IJCNN), M. Dang, P. Khosravi, Y. Liang, A. Vergari, and G. Van den Broeck (2021), Juice: a julia package for logic and probabilistic circuits, Proceedings of the 35th AAAI Conference on Artificial Intelligence (Demo Track), M. Dang, A. Vergari, and G. Broeck (2020), Strudel: learning structured-decomposable probabilistic circuits, The mnist database of handwritten digit images for machine learning research, L. Dinh, D. Krueger, and Y. Bengio (2014), Nice: non-linear independent components estimation, L. Dinh, J. Sohl-Dickstein, and S. Bengio (2016), International Conference on Learning Representations, Asymmetric numeral systems: entropy coding combining speed of huffman coding with compression rate of arithmetic coding, Learning the structure of sum-product networks, International conference on machine learning, I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio (2014), Advances in neural information processing systems, Keeping the neural networks simple by minimizing the description length of the weights, Proceedings of the sixth annual conference on Computational learning theory, Compression with flows via local bits-back coding, Proceedings of the 33rd International Conference on Neural Information Processing Systems, E. Hoogeboom, J. Peters, R. van den Berg, and M. Welling (2019), Integer discrete flows and lossless compression, Advances in Neural Information Processing Systems, A method for the construction of minimum-redundancy codes, Proceedings of the 32nd International Conference on Neural Information Processing Systems, D. P. Kingma, T. Salimans, R. Jozefowicz, X. Chen, I. Sutskever, and M. Welling (2016), Improved variational inference with inverse autoregressive flow, Proceedings of the 30th International Conference on Neural Information Processing Systems, Bit-swap: recursive bits-back coding for lossless compression with hierarchical latent variables, International Conference on Machine Learning, D. Kisa, G. Van den Broeck, A. Choi, and A. Darwiche (2014), Probabilistic sentential decision diagrams, Proceedings of the 14th international conference on principles of knowledge representation and reasoning (KR), Probabilistic graphical models: principles and techniques, Y. Lecun, S. Chopra, R. Hadsell, M. A. Ranzato, and F. J. Huang (2006), Y. Liang, J. Bekker, and G. Van den Broeck (2017), Learning the structure of probabilistic sentential decision diagrams, Proceedings of the 33rd Conference on Uncertainty in Artificial Intelligence (UAI), Tractable regularization of probabilistic circuits, Advances in Neural Information Processing Systems 35 (NeurIPS), F. Mentzer, E. Agustsson, M. Tschannen, R. Timofte, and L. V. Gool (2019), Practical full resolution learned lossless image compression, Proceedings of the IEEE/CVF conference on computer vision and pattern recognition. Finally, AC picks a number within the final interval that has the shortest binary representation. We have, where (a) reorders the terms for summation; (b) holds since nprod(\p,v\p), \pn(\x)=o\ch(n)\po(\x) and o\ch(n) such that {Xj}ij=1(o)=, \po(\x)=1;888This is because the scope of these PC units does not contain any of the variables in {Xj}ij=1. 8 (2016): 3061-3070. Since the PC is assumed to be balanced Sec. By scaling up the traditional PC structure learning pipeline, we achieve state-of-the-art results on image datasets such as MNIST. Lossless Compression with Probabilistic Circuits Anji Liu, Stephan Mandt, Guy Van den Broeck April, 2022 PDF Cite Code Abstract Despite extensive progress on image generation, common deep generative model architectures are not easily applied to lossless compression. This work proposes an algorithm to search for discrimination patterns in a more general class of probabilistic modelsprobabilistic circuitsand introduces new classes of discrimination patterns: minimal, maximal, and Pareto optimal. Semantic Scholar is a free, AI-powered research tool for scientific literature, based at the Allen Institute for AI. In addition, we have provided detailed algorithm tables Alg. Achieving optimal. Despite extensive progress on image generation, common deep generative model architectures are not easily applied to lossless compression. 6.4.4.1 Loss and Lossless Compression and Decompression Lossless compression is a compression technique that does not lose any data in the compression process. 3.2 may seem restrictive at first glance, we highlight that most existing PC structures such as EiNets (Peharz et al., 2020a), RAT-SPNs (Peharz et al., 2020b) and HCLTs (Sec. According to the variable order , vr,i uniquely exist for any i[D]. The algorithm adopts an CPU-based entropy coder rANS. Next, for x3. For a smooth structured-decomposable PC \p over D variables, for any scope , denote nodes(\p,) as the set of PC units in \p whose scope is . is the most widely used lossless . Next, PC+IDF improved over its base model IDF by 0.04, 0.16, and 0.19 bpd on three datasets, respectively. Product units of the rearranged omni-compatible circuits encoding p(X1) p(X2) p(X3) are shown in (c) and color-coded with those of matching scope in (a) and (b). Next, PCs can be integrated with Flow-/VAE-based compression methods. 2. Next, we justify the correctness of Alg. Since each vtree node corresponds to \bigO(\abs\pD) PC units, we need to evaluate \bigO(log(D)\abs\p) PC units to compute F(\x). This shows the benefit of integrating PCs with IDFs. Our results highlight the potential impact that non-standard learning architectures may have on neural data compression. This allows us to find a minimal set of models that provides an accurate description . Next, consider the inductive case where v is an inner node that has x descendent leaf nodes. 1. 1. Recent advances of deep generative models Figure 1. We use specialized activation and propagation methods to reduce pattern repetitions. Furthermore, a bigger PC model (M=32) with 7M parameters achieved codeword bpd 1.24, and is still 5x faster than BitSwap and IDF. At iteration i (\iewe want to compute the ith term in F(\x): \p(x1,,xi)), we need to evaluate all PC units that conform to any vtree node in the set T\p,i. Note that we use the full data when constructing/learning the PC. In the main loop (lines 5-6), the D terms in F(\x) are computed one-by-one. To encode xi, the algorithm partitions the current interval [a,b) using the left and right side cumulative probability of xi: Specifically, the algorithm updates [a,b) to the following: [a+(ba)li(xi),a+(ba)hi(xi)), which is a sub-interval of [a,b). Latent variable models such as VAEs rely on rate estimates obtained by lower-bounding the likelihood of the data, i.e., the quantity which is theoretically optimal for lossless compression; they furthermore rely on sophisticated schemes such as bits-back coding (Hinton and Van Camp, 1993) to realize these rates, oftentimes resulting in poor single-sample compression ratios (Kingma et al., 2019). On the other hand, models that make strong independence assumptions (\egn-gram, fully-factorized) are cheap to evaluate but lack the expressiveness to model complex distributions over structured data such as images.222 Back to our example, using the top-down probabilities, we can compute \p(X1=x1)=2i=1\pdown(ni)\pni(x1) without explicitly evaluating the ancestors of n1 and n2. Another way to identify these non-ancestor units is by inspecting their variable scopes if the variable scope of a PC unit n does not contain X1, it must has probability 1 while computing \p(X1=x1). (2020a), we use the gradients to compute the EM target of the parameters and performed mini-batch EM updates. 7(a) conforms to the vtree illustrated in Fig. Readme Stars. Notation We denote random variables by uppercase letters (\egX) and their assignments by lowercase letters (\egx). 3) based on Juice.jl (Dang et al., 2021), an open-source Julia package. In the following, we overview the empirical and theoretical results of the proposed (de)compression algorithm. For example, by using a small VAE model, Townsend et al. Languages. 1, we make use of another property, structured decomposability, which is the key to guaranteeing computational efficiency of the proposed (de)compression algorithm. This is achieved by targeting data that is deemed to be less noticeable so that the file itself still largely resembles the original. This work designs a predictive layer for structured-output prediction that can be plugged into any neural network guaranteeing its predictions are consistent with a set of symbolic constraints, and shows SPLs outperform competitors in terms of accuracy on challenging SOP tasks including hierarchical multi-label classication, path-to-end learning and preference learning, while retaining perfect constraint satisfaction. Capon, J. 3 is a detailed version of Alg. Compression and decompression with the PC+IDF model can be done easily: we can adopt the high-level compression algorithm of IDF and replace the parts of en- or decoding latent variables \zi with the proposed PC (de)compressor. Probabilistic Classification Using Elemental Abundance Distributions and Lossless Image Compression in Apollo 17 Lunar Dust Samples from Mare Serenitatis We have previously outlined a strategy for the detection of fossils [Storrie-Lombardi and Hoover, 2004] and extant microbial life [Storrie-Lombaudi and Hoover, 20051 during robotic missions to Mars using co-registered structural and chemical . See Sec. Circuit elements such as resistance, inductance, conductance, and capacitance (R, L, G, C) are referred to as primary line constants and are dependent on the geometry of the transmission line. Since n1 and n2 have the same variable scope, their distributions will not be multiplied by any product node. This study focuses specifically on lossless compression. As a key sub-routine in the proposed algorithm, we describe how to compute marginal queries given a smooth and (structured-)decomposable PC in \bigO(\abs\p) time. RocketChat Desktop via a Uniform Coder, DeepCABAC: A Universal Compression Algorithm for Deep Neural Networks, Practical Lossless Compression with Latent Variables using Bits Back Specifically, we demonstrate (i) how to select the set of PC units evali (\cfAlg. Although there exist various close-to-optimal compression schemes (\egHuffman Coding (Huffman, 1952) and Arithmetic Coding (Rissanen, 1976)), a natural question to ask is what are the requirements on the model \p such that compression algorithms can utilize it for encoding/decoding in a computationally efficient manner? Application of mesh and nodal analysis to circuits. With series resistance and series inductance, the transmission line is lossy. 1. As hinted by previous sections, PCs can be naturally integrated with existing neural compression algorithms: the simple latent variable distributions used by Flow- and VAE-based lossless compression methods can be replaced by more expressive distributions represented by PCs. it does not work well on random messages. These are a class of neural networks involving $|p|$ computational units that support efficient marginalization over arbitrary subsets of the $D$ feature dimensions, enabling efficient arithmetic coding. [Probabilistic Circuits] IDF was chosen because its authors provided an open-source implementation on GitHub. For example, each purple product unit (whose scope is {X1,X2}) has two children with disjoint scopes {X1} and {X2}, respectively. compared to available implementations of neural lossless compressors with near SoTA performance on datasets such as MNIST.111Note that there exists compression algorithms optimized particularly for speed by using simple entropy models (Townsend et al., 2019), though that also leads to worse bitrates. which is a property of the (variable) scope (n) of PC units n, that is, the collection of variables defined by all its descendent input units. No packages published . Although recent breakthroughs have led to PCs that can generate CelebA and SVHN images (Peharz et al., 2020a), PCs have not been shown to have competitive (normalized) likelihoods on image datasets, which directly influence compression rates. |p|), where a naive scheme would have linear costs As shown in Table 2, the PC (de)compressor achieved compression rates close to its theoretical rate estimate codeword bpds only have \vbox0.04 loss \wrtthe corresponding theoretical bpds. Analogously, we use bold uppercase (\eg\X) and lowercase (\eg\x) letters to denote sets of variables and their joint assignments, respectively. 7(c); the corresponding PC (Fig. Competitive runtimes. To overcome such problems, we establish a new class of tractable lossless compression models that permit efficient encoding and decoding: Probabilistic Circuits (PCs). Note that this operation will not change the number of parameters in a PC, and will only incur at most 2\abs\p edges. that non-standard learning architectures may have on neural data compression. Practical (neural) lossless compression algorithms operate in two main phases learning and compression (Yang et al., 2022). Lossless compression does not result in losing this data, but at the cost of a smaller reduction in file size. All experiments were performed using the following official code: https://github.com/bits-back/bits-back. Furthermore, PCs can be naturally integrated with existing neural compression algorithms to improve the performance of these base models on natural image datasets. It is very important that the reconstruction is identical to the text original, as very small . A new coding scheme. For a pair of ordered PC and ordered vtree, the optimal variable order is defined as the order the leaf vtree nodes (each corresponds to a variable) are accessed following an inorder traverse of the vtree (left child accessed before right child). SoTA likelihood on MNIST. Abstract: Despite extensive progress on image generation, deep generative models are suboptimal when applied to lossless compression. The quantity \pdown(n) of all PC units n can be computed by Alg. The goal of lossless image compression is to represent an image signal with the smallest possible number of bits without loss of any information, thereby speeding up transmission and minimizing storage requirements. In theory, PC can be integrated with any VAE- and Flow-based model. As a result, it can cause some degradation that reduces the image quality. This introduction hopes to fill in the necessary background by reviewing basic coding topics such as entropy coding and rate-distortion theory, related machine learning ideas such as bits-back coding and perceptual metrics, and providing a guide through the representative works in the literature so far. We can significantly accelerate the en- and decoding times if the PC is structured-decomposable (see Definition1). Client, Lossless Compression with Probabilistic Circuits, This website places cookies on your device to give you the best user experience. Lossless Image Compression. However, since enforcing smoothness on any structured-decomposable PC only imposes at most an almost-linear increase in its size (Shih et al., 2019), we omit introducing it here (all PCs used in this paper are structured-decomposable). . Therefore, the main computation cost of the above en- and decoding procedures comes from calculating the 2D conditional probabilities {li(x),hi(x)}Di=1 \wrtany \x. We adopt two types of EM updates mini-batch and full-batch. 7(d)). While Sec. 1 are decomposable. Full-batch EM updates the parameters with \params(EM) at each iteration. 7 Compression Technologies Statistical redundancy Lossless compression Also known as entropy coding Build on the probabilistic characteristics of signals Perceptual redundancy Lossy compression Lead to irreversible distortion Complex and depends on context or applications First, every PC unit with scope {X1} (\iethe two nodes colored blue) has to be evaluated. Therefore, most of the image compression techniques presented are lossless compression [10-16]. For example, Arithmetic Coding (AC) encodes the symbols {xi}Di=1 (define D:=\abs\X as the number of features) sequentially by successively refining an interval that represents the sample, starting from the initial interval [0,1). The file can be decompressed to its original quality without any loss of data. While both Flow- and VAE-based compression algorithms (Hoogeboom et al., 2019; Kingma et al., 2019) support efcient and near-optimal compression under certain assumptions (e.g., the existence of an additional source of random bits), we show that Probabilistic Circuits (PCs) (Choi et al., 2020) are also suitable for lossless compression tasks. 1? 2; if evidence of X is not given, probability 1 is assigned to n. Next, we do a feedforward (children before parents) traverse of inner PC units and compute their probabilities following Eq. A. Gritsenko, M. Dehghani, C. K. Snderby, and T. Salimans (2020), Idf++: analyzing and improving integer discrete flows for lossless compression, H. Xiao, K. Rasul, and R. Vollgraf (2017), Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, An introduction to neural data compression, M. Zhang, A. Zhang, and S. McDonagh (2021a), On the out-of-distribution generalization of probabilistic image modelling, S. Zhang, C. Zhang, N. Kang, and Z. Li (2021b), iVPF: numerical invertible volume preserving flow for efficient lossless compression, Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, Disentangling latent space for vae by label relevant/irrelevant dimensions, the~{}top-down~{}probability~{}of~{}every~{}% In mini-batch EM, parameters are updated according to a step size : \params(k+1)(1)\params(k)+\params% Once the loss has converged, we have a K K K-parameter model that allows us to cluster or label new, unseen data.For example, once the model is trained on Old Faithful data as in Figure 1, we can assign new Old Faithful data to one of the two clusters. Next lesson. 3, is an algorithm that computes the 2D conditional probabilities {li(x),hi(x)}Di=1 given any \x efficiently, as justified by the following theorem. We proceed to evaluate the generative performance of the proposed PC+IDF model on 3 natural image datasets: CIFAR10, ImageNet32, and ImageNet64. Upon decoding, the symbols {xi}Di=1 are decoded sequentially: at iteration i, we decode variable Xi by looking up its value x such that its cumulative probability (\ieli(x)) matches the subinterval specified by the codeword and x1,,xi1 (Rissanen, 1976); the decoded symbol xi is then used to compute the following conditional probabilities (\ielj(x) for j>i). This procedure is formalized in Alg. The kernels automatically segment PC units into layers such that the computation in every layer can be fully parallelized. Lossless compression is a class of data compression that allows the original data to be perfectly reconstructed from the compressed data with no loss of information. 309 21. Full convolutional VAE models trained on 32x32 ImageNet can generalize well, not just to 64x64 but also to far larger photographs, with no changes to the model, achieving state of the art for compression of full size ImageNet images. In this section, we show that Hidden Chow-Liu Trees (HCLTs) (Liu and Van den Broeck, 2021), a PC model initially proposed for simple density estimation tasks containing binary features, can be scaled up to achieve state-of-the-art performance on various image datasets. Introduction to Laplace . SoTA likelihood on MNIST. 1 is structured-decomposable because for all three groups of product units with the same scope (grouped by their colors), their children divide the variable scope in the same way. Sign up to our mailing list for occasional updates. See Sec. Lossless compression is generally used for so-called "discrete" data, such as database records, spreadsheets, word-processing files, and even some kinds of image and video information. Data Compression With Deep Probabilistic Models. To ensure that a PC models a valid distribution, we assume the parameters associated with any sum unit n are normalized: n,c\ch(n)n,c=1. Since both IDF and the PC models are fully differentiable, the PC+IDF model can be trained end-to-end via gradient descent. A PC is structured-decomposable if (i) it is decomposable and (ii) for every pair of product units (m,n) with identical scope (\ie(m)=(n)), we have that \abs\ch(m)=\abs\ch(n) and the scopes of their children are pairwise identical: i{1,,\abs\ch(m)},(cmi)=(cni), where cmi and cni are the ith child unit of m and n. The PC shown in Fig. As shown in Table 1, the small PC model achieved a near-SoTA bitrate while being \vbox15x faster Flow-model-based neural compression algorithms adopt \p defined on mutually independent latent variables (denoted \Z), and improve expressiveness by learning bijection functions between \Z and \X (\iethe input space). The integrated model is illustrated in Fig. For a smooth structured-decomposable balanced PC \p over D variables \X and a sample \x, there exists a variable order , s.t. We ran all experiments with the code in the GitHub repo provided by the authors. Add a Instead, we propose to model every set of latent variables \zi with a PC \p(\zi). To improve expressiveness, latent variables are added to the CLT by the two following steps: (i) replace observed variables Xi by their corresponding latent variables Zi, which are defined to be categorical variables with M (a hyperparameter) categories (Fig. For example, models such as VAEs suffer from a compression. Let c1 and c2 have y and z descendent leaf nodes, respectively. We achieve this by recognizing three types of PC units that can be safely eliminated for evaluation. 7(b) is transformed into an ordered vtree illustrated in Fig. 3). Practice: Lossy vs. lossless compression. Since there are in total (D) different variable scopes in \p, we have: for any scope exists in an HCLT \p, nodes(\p,)=\bigO(\abs\p/D). (2019)) neural compression algorithms using the MNIST dataset. 3. Unlike lossy compression, lossless image compression won't reduce image quality. Consider compressing an 2-by-2 image, whose four pixels are denoted as X1,,X4. This work follows a simple deep learning approach, by generating unspecialized random structures, scalable to millions of parameters, and subsequently applying GPUbased optimization, which yields well-calibrated uncertainties and stands out among most deep generative and discriminative models in being robust to missing features and being able to detect anomalies. [1] Specifically, leaf PGM nodes corresponding to observed variables Xi are compiled into PC input units of Xi (line 6), and inner PGM nodes corresponding to latent variables are compiled by taking products and sums (implemented by product and sum units) of its child nodes PC units (lines 8-10). stardew valley character B.4 for more details about the experiments. They provide a set of succinct definitions for popular TPMs such as Sum-Product Networks (Poon and Domingos, 2011), Arithmetic Circuits (Shen et al., 2016), and Probabilistic Sentential Decision Diagrams (Kisa et al., 2014). LOSSLESS COMPRESSION ALGORITHMS FOR POST-OPC IC LAYOUT Allan Gu and Avideh Zakhor Department of Electrical Engineering and Computer Sciences University of California at Berkeley, CA 94720, USA ABSTRACT An important step in today's Integrated Circuit (IC) manu- facturing is optical proximity correction (OPC). As a result, the length of b is approximately log\p(\x). Each level contains a squeeze layer (Dinh et al., 2016), followed by several integer flow layers and a prior layer. Next, every PC unit n that is not an ancestor of the two blue units (\ienon-ancestor units in Fig. During mini-batch EM updates, was annealed linearly from 0.15 to 0.05. For example, VAEs suffer from a compression cost overhead due to their latent variables. Specially, we decompose the in- put image x into the lossy reconstruction xlproduced by BPG and the corresponding residual r. Specifically, Fig. In our experiments, we trained the HCLTs with 100 mini-batch EM epochs and 20 full-batch EM epochs. We derive efficient encoding and decoding schemes that both have time complexity O(log(D)|p|), where a naive scheme would have linear costs in D and |p|, making the approach highly scalable. In addition to Fig. Lossless compression using Probabilistic Circuits. As hinted by Sec. . All product units in Fig. Specifically, we ask: are there probabilistic models that are both expressive and permit efficient computation of the conditional probabilities in Eq. Specifically, the original image is first decomposed into the lossy reconstruction obtained after compressing it with BPG and the corresponding residual. 3(c), which is detailed in Sec. This is different from regular neural networks and It is also completely reversible. LLCC operates in a similar fashion to the existing clear channel codec: the decoded 64kbps PCM stream is a bit-exact replica of the PCM stream provided on the TDM side of the encoding DSP. 5, we aim to provide an intuitive illustration to both problems. By scaling up the traditional PC All 6 methods were tested on 4 datasets, which include MNIST (Deng, 2012), FashionMNIST (Xiao et al., 2017), and two splits of EMNIST (Cohen et al., 2017). 3.1, and the backbone inference algorithm with \bigO(log(D)\abs\p) time complexity will later be shown as Alg. Abstract and Figures. The high-level idea of the proof is to first show how to compute the optimal variable order for any smooth and structured-decomposable PC. 1),333Another property called smoothness is also required to compute marginals efficiently. arXiv Vanity renders academic papers from 1. For example, the PC shown in Fig. Details regarding model architecture and parameter learning are provided in Sec. The advantage of the proposed binomial adaptive compression method compared with the binomial compression method previously developed by the authors is an increase in the compression rate. As mentioned in the main text, computing the pairwise mutual information between variables \X is the first step to compute the Chow-Liu Tree. As a result, we only evaluate 20 PC units in total, compared to 3\abs\p=39 units required by the naive approach. The compressors find the probability of the next character based on multiple human-designed contexts/features (eg: past 20 chars, 4 words, or alternate characters, only the higher bits of the last 10 bytes etc.). To encode a sample \x, a standard streaming code operates by sequentially encoding every symbol xi into a bitstream b, such that xi occupies approximately log\p(xi|x1,,xi1) bits in b. While both Flow- and VAE-based compression algorithms (Hoogeboom et al., 2019; Kingma et al., 2019) support efficient and near-optimal compression under certain assumptions (\egthe existence of an additional source of random bits), we show that Probabilistic Circuits (PCs) (Choi et al., 2020) are also suitable for lossless compression tasks. While computing each term, we first find the PC units that need to be evaluated (line 6).77footnotemark: 7After computing their probabilities in a bottom-up manner (line 7), we additionally use the pre-computed top-down probabilities to obtain the target marginal probability (lines 8-9).
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