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1. Sample Complexity Analysis and Self-regularization in Identification of Over-parameterized ARX Models

   Du, Zhe ; Liu, Zexiang ; Weitz, Jack ; Ozay, Necmiye ( January 2022 , IEEE Conference on Decision and Control (CDC))

   AutoRegressive eXogenous (ARX) models form one of the most important model classes in control theory, econometrics, and statistics, but they are yet to be understood in terms of their finite sample identification analysis. The technical challenges come from the strong statistical dependency not only between data samples at different time instances but also between elements within each individual sample. In this work, for ARX models with potentially unknown orders, we study how ordinary least squares (OLS) estimator performs in terms of identifying model parameters from data collected from either a single length-T trajectory or N i.i.d. trajectories. Our main results show that as long as the orders of the model are chosen optimistically, i.e., we are learning an over-parameterized model compared to the ground truth ARX, the OLS will converge with the optimal rate O(1/√T) (or O(1/√N)) to the true (low-order) ARX parameters. This occurs without the aid of any regularization, thus is referred to as self-regularization. Our results imply that the oracle knowledge of the true orders and usage of regularizers are not necessary in learning ARX models — over-parameterization is all you need 

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2. The Temple University Digital Pathology Corpus: The Breast Tissue Subset

   https://doi.org/10.1109/SPMB52430.2021.9672275  

   Wevodau, Z. ; Doshna, B. ; Jhala, N. ; Akhtar, I. ; Obeid, I. ; Picone, J. ( December 2021 , Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB))

   Obeid, I. (Ed.)

   The Neural Engineering Data Consortium (NEDC) is developing the Temple University Digital Pathology Corpus (TUDP), an open source database of high-resolution images from scanned pathology samples [1], as part of its National Science Foundation-funded Major Research Instrumentation grant titled “MRI: High Performance Digital Pathology Using Big Data and Machine Learning” [2]. The long-term goal of this project is to release one million images. We have currently scanned over 100,000 images and are in the process of annotating breast tissue data for our first official corpus release, v1.0.0. This release contains 3,505 annotated images of breast tissue including 74 patients with cancerous diagnoses (out of a total of 296 patients). In this poster, we will present an analysis of this corpus and discuss the challenges we have faced in efficiently producing high quality annotations of breast tissue. It is well known that state of the art algorithms in machine learning require vast amounts of data. Fields such as speech recognition [3], image recognition [4] and text processing [5] are able to deliver impressive performance with complex deep learning models because they have developed large corpora to support training of extremely high-dimensional models (e.g., billions of parameters). Other fields that do not have access to such data resources must rely on techniques in which existing models can be adapted to new datasets [6]. A preliminary version of this breast corpus release was tested in a pilot study using a baseline machine learning system, ResNet18 [7], that leverages several open-source Python tools. The pilot corpus was divided into three sets: train, development, and evaluation. Portions of these slides were manually annotated [1] using the nine labels in Table 1 [8] to identify five to ten examples of pathological features on each slide. Not every pathological feature is annotated, meaning excluded areas can include focuses particular to these labels that are not used for training. A summary of the number of patches within each label is given in Table 2. To maintain a balanced training set, 1,000 patches of each label were used to train the machine learning model. Throughout all sets, only annotated patches were involved in model development. The performance of this model in identifying all the patches in the evaluation set can be seen in the confusion matrix of classification accuracy in Table 3. The highest performing labels were background, 97% correct identification, and artifact, 76% correct identification. A correlation exists between labels with more than 6,000 development patches and accurate performance on the evaluation set. Additionally, these results indicated a need to further refine the annotation of invasive ductal carcinoma (“indc”), inflammation (“infl”), nonneoplastic features (“nneo”), normal (“norm”) and suspicious (“susp”). This pilot experiment motivated changes to the corpus that will be discussed in detail in this poster presentation. To increase the accuracy of the machine learning model, we modified how we addressed underperforming labels. One common source of error arose with how non-background labels were converted into patches. Large areas of background within other labels were isolated within a patch resulting in connective tissue misrepresenting a non-background label. In response, the annotation overlay margins were revised to exclude benign connective tissue in non-background labels. Corresponding patient reports and supporting immunohistochemical stains further guided annotation reviews. The microscopic diagnoses given by the primary pathologist in these reports detail the pathological findings within each tissue site, but not within each specific slide. The microscopic diagnoses informed revisions specifically targeting annotated regions classified as cancerous, ensuring that the labels “indc” and “dcis” were used only in situations where a micropathologist diagnosed it as such. Further differentiation of cancerous and precancerous labels, as well as the location of their focus on a slide, could be accomplished with supplemental immunohistochemically (IHC) stained slides. When distinguishing whether a focus is a nonneoplastic feature versus a cancerous growth, pathologists employ antigen targeting stains to the tissue in question to confirm the diagnosis. For example, a nonneoplastic feature of usual ductal hyperplasia will display diffuse staining for cytokeratin 5 (CK5) and no diffuse staining for estrogen receptor (ER), while a cancerous growth of ductal carcinoma in situ will have negative or focally positive staining for CK5 and diffuse staining for ER [9]. Many tissue samples contain cancerous and non-cancerous features with morphological overlaps that cause variability between annotators. The informative fields IHC slides provide could play an integral role in machine model pathology diagnostics. Following the revisions made on all the annotations, a second experiment was run using ResNet18. Compared to the pilot study, an increase of model prediction accuracy was seen for the labels indc, infl, nneo, norm, and null. This increase is correlated with an increase in annotated area and annotation accuracy. Model performance in identifying the suspicious label decreased by 25% due to the decrease of 57% in the total annotated area described by this label. A summary of the model performance is given in Table 4, which shows the new prediction accuracy and the absolute change in error rate compared to Table 3. The breast tissue subset we are developing includes 3,505 annotated breast pathology slides from 296 patients. The average size of a scanned SVS file is 363 MB. The annotations are stored in an XML format. A CSV version of the annotation file is also available which provides a flat, or simple, annotation that is easy for machine learning researchers to access and interface to their systems. Each patient is identified by an anonymized medical reference number. Within each patient’s directory, one or more sessions are identified, also anonymized to the first of the month in which the sample was taken. These sessions are broken into groupings of tissue taken on that date (in this case, breast tissue). A deidentified patient report stored as a flat text file is also available. Within these slides there are a total of 16,971 total annotated regions with an average of 4.84 annotations per slide. Among those annotations, 8,035 are non-cancerous (normal, background, null, and artifact,) 6,222 are carcinogenic signs (inflammation, nonneoplastic and suspicious,) and 2,714 are cancerous labels (ductal carcinoma in situ and invasive ductal carcinoma in situ.) The individual patients are split up into three sets: train, development, and evaluation. Of the 74 cancerous patients, 20 were allotted for both the development and evaluation sets, while the remain 34 were allotted for train. The remaining 222 patients were split up to preserve the overall distribution of labels within the corpus. This was done in hope of creating control sets for comparable studies. Overall, the development and evaluation sets each have 80 patients, while the training set has 136 patients. In a related component of this project, slides from the Fox Chase Cancer Center (FCCC) Biosample Repository (https://www.foxchase.org/research/facilities/genetic-research-facilities/biosample-repository -facility) are being digitized in addition to slides provided by Temple University Hospital. This data includes 18 different types of tissue including approximately 38.5% urinary tissue and 16.5% gynecological tissue. These slides and the metadata provided with them are already anonymized and include diagnoses in a spreadsheet with sample and patient ID. We plan to release over 13,000 unannotated slides from the FCCC Corpus simultaneously with v1.0.0 of TUDP. Details of this release will also be discussed in this poster. Few digitally annotated databases of pathology samples like TUDP exist due to the extensive data collection and processing required. The breast corpus subset should be released by November 2021. By December 2021 we should also release the unannotated FCCC data. We are currently annotating urinary tract data as well. We expect to release about 5,600 processed TUH slides in this subset. We have an additional 53,000 unprocessed TUH slides digitized. Corpora of this size will stimulate the development of a new generation of deep learning technology. In clinical settings where resources are limited, an assistive diagnoses model could support pathologists’ workload and even help prioritize suspected cancerous cases. ACKNOWLEDGMENTS This material is supported by the National Science Foundation under grants nos. CNS-1726188 and 1925494. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. REFERENCES [1] N. Shawki et al., “The Temple University Digital Pathology Corpus,” in Signal Processing in Medicine and Biology: Emerging Trends in Research and Applications, 1st ed., I. Obeid, I. Selesnick, and J. Picone, Eds. New York City, New York, USA: Springer, 2020, pp. 67 104. https://www.springer.com/gp/book/9783030368432. [2] J. Picone, T. Farkas, I. Obeid, and Y. Persidsky, “MRI: High Performance Digital Pathology Using Big Data and Machine Learning.” Major Research Instrumentation (MRI), Division of Computer and Network Systems, Award No. 1726188, January 1, 2018 – December 31, 2021. https://www. isip.piconepress.com/projects/nsf_dpath/. [3] A. Gulati et al., “Conformer: Convolution-augmented Transformer for Speech Recognition,” in Proceedings of the Annual Conference of the International Speech Communication Association (INTERSPEECH), 2020, pp. 5036-5040. https://doi.org/10.21437/interspeech.2020-3015. [4] C.-J. Wu et al., “Machine Learning at Facebook: Understanding Inference at the Edge,” in Proceedings of the IEEE International Symposium on High Performance Computer Architecture (HPCA), 2019, pp. 331–344. https://ieeexplore.ieee.org/document/8675201. [5] I. Caswell and B. Liang, “Recent Advances in Google Translate,” Google AI Blog: The latest from Google Research, 2020. [Online]. Available: https://ai.googleblog.com/2020/06/recent-advances-in-google-translate.html. [Accessed: 01-Aug-2021]. [6] V. Khalkhali, N. Shawki, V. Shah, M. Golmohammadi, I. Obeid, and J. Picone, “Low Latency Real-Time Seizure Detection Using Transfer Deep Learning,” in Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2021, pp. 1 7. https://www.isip. piconepress.com/publications/conference_proceedings/2021/ieee_spmb/eeg_transfer_learning/. [7] J. Picone, T. Farkas, I. Obeid, and Y. Persidsky, “MRI: High Performance Digital Pathology Using Big Data and Machine Learning,” Philadelphia, Pennsylvania, USA, 2020. https://www.isip.piconepress.com/publications/reports/2020/nsf/mri_dpath/. [8] I. Hunt, S. Husain, J. Simons, I. Obeid, and J. Picone, “Recent Advances in the Temple University Digital Pathology Corpus,” in Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB), 2019, pp. 1–4. https://ieeexplore.ieee.org/document/9037859. [9] A. P. Martinez, C. Cohen, K. Z. Hanley, and X. (Bill) Li, “Estrogen Receptor and Cytokeratin 5 Are Reliable Markers to Separate Usual Ductal Hyperplasia From Atypical Ductal Hyperplasia and Low-Grade Ductal Carcinoma In Situ,” Arch. Pathol. Lab. Med., vol. 140, no. 7, pp. 686–689, Apr. 2016. https://doi.org/10.5858/arpa.2015-0238-OA. 

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3. A framework for machine learning of model error in dynamical systems

   https://doi.org/10.1090/cams/10  

   Levine, Matthew ; Stuart, Andrew ( October 2022 , Communications of the American Mathematical Society)

   The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems from noisily and partially observed data. We compare pure data-driven learning with hybrid models which incorporate imperfect domain knowledge, referring to the discrepancy between an assumed truth model and the imperfect mechanistic model as model error. Our formulation is agnostic to the chosen machine learning model, is presented in both continuous- and discrete-time settings, and is compatible both with model errors that exhibit substantial memory and errors that are memoryless. First, we study memoryless linear (w.r.t. parametric-dependence) model error from a learning theory perspective, defining excess risk and generalization error. For ergodic continuous-time systems, we prove that both excess risk and generalization error are bounded above by terms that diminish with the square-root of T T , the time-interval over which training data is specified. Secondly, we study scenarios that benefit from modeling with memory, proving universal approximation theorems for two classes of continuous-time recurrent neural networks (RNNs): both can learn memory-dependent model error, assuming that it is governed by a finite-dimensional hidden variable and that, together, the observed and hidden variables form a continuous-time Markovian system. In addition, we connect one class of RNNs to reservoir computing, thereby relating learning of memory-dependent error to recent work on supervised learning between Banach spaces using random features. Numerical results are presented (Lorenz ’63, Lorenz ’96 Multiscale systems) to compare purely data-driven and hybrid approaches, finding hybrid methods less datahungry and more parametrically efficient. We also find that, while a continuous-time framing allows for robustness to irregular sampling and desirable domain- interpretability, a discrete-time framing can provide similar or better predictive performance, especially when data are undersampled and the vector field defining the true dynamics cannot be identified. Finally, we demonstrate numerically how data assimilation can be leveraged to learn hidden dynamics from noisy, partially-observed data, and illustrate challenges in representing memory by this approach, and in the training of such models. 

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4. A Deep Learning-Based Real-time Seizure Detection System

   https://doi.org/10.1109/SPMB50085.2020.9353623  

   Shawki, N. ; Elseify, T. ; Cap, T. ; Shah, V. ; Obeid, I. ; Picone, J. ( December 2020 , Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium (SPMB))

   Obeid, Iyad Selesnick (Ed.)

   Electroencephalography (EEG) is a popular clinical monitoring tool used for diagnosing brain-related disorders such as epilepsy [1]. As monitoring EEGs in a critical-care setting is an expensive and tedious task, there is a great interest in developing real-time EEG monitoring tools to improve patient care quality and efficiency [2]. However, clinicians require automatic seizure detection tools that provide decisions with at least 75% sensitivity and less than 1 false alarm (FA) per 24 hours [3]. Some commercial tools recently claim to reach such performance levels, including the Olympic Brainz Monitor [4] and Persyst 14 [5]. In this abstract, we describe our efforts to transform a high-performance offline seizure detection system [3] into a low latency real-time or online seizure detection system. An overview of the system is shown in Figure 1. The main difference between an online versus offline system is that an online system should always be causal and has minimum latency which is often defined by domain experts. The offline system, shown in Figure 2, uses two phases of deep learning models with postprocessing [3]. The channel-based long short term memory (LSTM) model (Phase 1 or P1) processes linear frequency cepstral coefficients (LFCC) [6] features from each EEG channel separately. We use the hypotheses generated by the P1 model and create additional features that carry information about the detected events and their confidence. The P2 model uses these additional features and the LFCC features to learn the temporal and spatial aspects of the EEG signals using a hybrid convolutional neural network (CNN) and LSTM model. Finally, Phase 3 aggregates the results from both P1 and P2 before applying a final postprocessing step. The online system implements Phase 1 by taking advantage of the Linux piping mechanism, multithreading techniques, and multi-core processors. To convert Phase 1 into an online system, we divide the system into five major modules: signal preprocessor, feature extractor, event decoder, postprocessor, and visualizer. The system reads 0.1-second frames from each EEG channel and sends them to the feature extractor and the visualizer. The feature extractor generates LFCC features in real time from the streaming EEG signal. Next, the system computes seizure and background probabilities using a channel-based LSTM model and applies a postprocessor to aggregate the detected events across channels. The system then displays the EEG signal and the decisions simultaneously using a visualization module. The online system uses C++, Python, TensorFlow, and PyQtGraph in its implementation. The online system accepts streamed EEG data sampled at 250 Hz as input. The system begins processing the EEG signal by applying a TCP montage [8]. Depending on the type of the montage, the EEG signal can have either 22 or 20 channels. To enable the online operation, we send 0.1-second (25 samples) length frames from each channel of the streamed EEG signal to the feature extractor and the visualizer. Feature extraction is performed sequentially on each channel. The signal preprocessor writes the sample frames into two streams to facilitate these modules. In the first stream, the feature extractor receives the signals using stdin. In parallel, as a second stream, the visualizer shares a user-defined file with the signal preprocessor. This user-defined file holds raw signal information as a buffer for the visualizer. The signal preprocessor writes into the file while the visualizer reads from it. Reading and writing into the same file poses a challenge. The visualizer can start reading while the signal preprocessor is writing into it. To resolve this issue, we utilize a file locking mechanism in the signal preprocessor and visualizer. Each of the processes temporarily locks the file, performs its operation, releases the lock, and tries to obtain the lock after a waiting period. The file locking mechanism ensures that only one process can access the file by prohibiting other processes from reading or writing while one process is modifying the file [9]. The feature extractor uses circular buffers to save 0.3 seconds or 75 samples from each channel for extracting 0.2-second or 50-sample long center-aligned windows. The module generates 8 absolute LFCC features where the zeroth cepstral coefficient is replaced by a temporal domain energy term. For extracting the rest of the features, three pipelines are used. The differential energy feature is calculated in a 0.9-second absolute feature window with a frame size of 0.1 seconds. The difference between the maximum and minimum temporal energy terms is calculated in this range. Then, the first derivative or the delta features are calculated using another 0.9-second window. Finally, the second derivative or delta-delta features are calculated using a 0.3-second window [6]. The differential energy for the delta-delta features is not included. In total, we extract 26 features from the raw sample windows which add 1.1 seconds of delay to the system. We used the Temple University Hospital Seizure Database (TUSZ) v1.2.1 for developing the online system [10]. The statistics for this dataset are shown in Table 1. A channel-based LSTM model was trained using the features derived from the train set using the online feature extractor module. A window-based normalization technique was applied to those features. In the offline model, we scale features by normalizing using the maximum absolute value of a channel [11] before applying a sliding window approach. Since the online system has access to a limited amount of data, we normalize based on the observed window. The model uses the feature vectors with a frame size of 1 second and a window size of 7 seconds. We evaluated the model using the offline P1 postprocessor to determine the efficacy of the delayed features and the window-based normalization technique. As shown by the results of experiments 1 and 4 in Table 2, these changes give us a comparable performance to the offline model. The online event decoder module utilizes this trained model for computing probabilities for the seizure and background classes. These posteriors are then postprocessed to remove spurious detections. The online postprocessor receives and saves 8 seconds of class posteriors in a buffer for further processing. It applies multiple heuristic filters (e.g., probability threshold) to make an overall decision by combining events across the channels. These filters evaluate the average confidence, the duration of a seizure, and the channels where the seizures were observed. The postprocessor delivers the label and confidence to the visualizer. The visualizer starts to display the signal as soon as it gets access to the signal file, as shown in Figure 1 using the “Signal File” and “Visualizer” blocks. Once the visualizer receives the label and confidence for the latest epoch from the postprocessor, it overlays the decision and color codes that epoch. The visualizer uses red for seizure with the label SEIZ and green for the background class with the label BCKG. Once the streaming finishes, the system saves three files: a signal file in which the sample frames are saved in the order they were streamed, a time segmented event (TSE) file with the overall decisions and confidences, and a hypotheses (HYP) file that saves the label and confidence for each epoch. The user can plot the signal and decisions using the signal and HYP files with only the visualizer by enabling appropriate options. For comparing the performance of different stages of development, we used the test set of TUSZ v1.2.1 database. It contains 1015 EEG records of varying duration. The any-overlap performance [12] of the overall system shown in Figure 2 is 40.29% sensitivity with 5.77 FAs per 24 hours. For comparison, the previous state-of-the-art model developed on this database performed at 30.71% sensitivity with 6.77 FAs per 24 hours [3]. The individual performances of the deep learning phases are as follows: Phase 1’s (P1) performance is 39.46% sensitivity and 11.62 FAs per 24 hours, and Phase 2 detects seizures with 41.16% sensitivity and 11.69 FAs per 24 hours. We trained an LSTM model with the delayed features and the window-based normalization technique for developing the online system. Using the offline decoder and postprocessor, the model performed at 36.23% sensitivity with 9.52 FAs per 24 hours. The trained model was then evaluated with the online modules. The current performance of the overall online system is 45.80% sensitivity with 28.14 FAs per 24 hours. Table 2 summarizes the performances of these systems. The performance of the online system deviates from the offline P1 model because the online postprocessor fails to combine the events as the seizure probability fluctuates during an event. The modules in the online system add a total of 11.1 seconds of delay for processing each second of the data, as shown in Figure 3. In practice, we also count the time for loading the model and starting the visualizer block. When we consider these facts, the system consumes 15 seconds to display the first hypothesis. The system detects seizure onsets with an average latency of 15 seconds. Implementing an automatic seizure detection model in real time is not trivial. We used a variety of techniques such as the file locking mechanism, multithreading, circular buffers, real-time event decoding, and signal-decision plotting to realize the system. A video demonstrating the system is available at: https://www.isip.piconepress.com/projects/nsf_pfi_tt/resources/videos/realtime_eeg_analysis/v2.5.1/video_2.5.1.mp4. The final conference submission will include a more detailed analysis of the online performance of each module. ACKNOWLEDGMENTS Research reported in this publication was most recently supported by the National Science Foundation Partnership for Innovation award number IIP-1827565 and the Pennsylvania Commonwealth Universal Research Enhancement Program (PA CURE). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the official views of any of these organizations. REFERENCES [1] A. Craik, Y. He, and J. L. Contreras-Vidal, “Deep learning for electroencephalogram (EEG) classification tasks: a review,” J. Neural Eng., vol. 16, no. 3, p. 031001, 2019. https://doi.org/10.1088/1741-2552/ab0ab5. [2] A. C. Bridi, T. Q. Louro, and R. C. L. Da Silva, “Clinical Alarms in intensive care: implications of alarm fatigue for the safety of patients,” Rev. Lat. Am. Enfermagem, vol. 22, no. 6, p. 1034, 2014. https://doi.org/10.1590/0104-1169.3488.2513. [3] M. Golmohammadi, V. Shah, I. Obeid, and J. Picone, “Deep Learning Approaches for Automatic Seizure Detection from Scalp Electroencephalograms,” in Signal Processing in Medicine and Biology: Emerging Trends in Research and Applications, 1st ed., I. Obeid, I. Selesnick, and J. Picone, Eds. New York, New York, USA: Springer, 2020, pp. 233–274. https://doi.org/10.1007/978-3-030-36844-9_8. [4] “CFM Olympic Brainz Monitor.” [Online]. Available: https://newborncare.natus.com/products-services/newborn-care-products/newborn-brain-injury/cfm-olympic-brainz-monitor. [Accessed: 17-Jul-2020]. [5] M. L. Scheuer, S. B. Wilson, A. Antony, G. Ghearing, A. Urban, and A. I. Bagic, “Seizure Detection: Interreader Agreement and Detection Algorithm Assessments Using a Large Dataset,” J. Clin. Neurophysiol., 2020. https://doi.org/10.1097/WNP.0000000000000709. [6] A. Harati, M. Golmohammadi, S. Lopez, I. Obeid, and J. Picone, “Improved EEG Event Classification Using Differential Energy,” in Proceedings of the IEEE Signal Processing in Medicine and Biology Symposium, 2015, pp. 1–4. https://doi.org/10.1109/SPMB.2015.7405421. [7] V. Shah, C. Campbell, I. Obeid, and J. Picone, “Improved Spatio-Temporal Modeling in Automated Seizure Detection using Channel-Dependent Posteriors,” Neurocomputing, 2021. [8] W. Tatum, A. Husain, S. Benbadis, and P. Kaplan, Handbook of EEG Interpretation. New York City, New York, USA: Demos Medical Publishing, 2007. [9] D. P. Bovet and C. Marco, Understanding the Linux Kernel, 3rd ed. O’Reilly Media, Inc., 2005. https://www.oreilly.com/library/view/understanding-the-linux/0596005652/. [10] V. Shah et al., “The Temple University Hospital Seizure Detection Corpus,” Front. Neuroinform., vol. 12, pp. 1–6, 2018. https://doi.org/10.3389/fninf.2018.00083. [11] F. Pedregosa et al., “Scikit-learn: Machine Learning in Python,” J. Mach. Learn. Res., vol. 12, pp. 2825–2830, 2011. https://dl.acm.org/doi/10.5555/1953048.2078195. [12] J. Gotman, D. Flanagan, J. Zhang, and B. Rosenblatt, “Automatic seizure detection in the newborn: Methods and initial evaluation,” Electroencephalogr. Clin. Neurophysiol., vol. 103, no. 3, pp. 356–362, 1997. https://doi.org/10.1016/S0013-4694(97)00003-9. 

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5. Local Access to Huge Random Objects Through Partial Sampling

   Biswas, Amartya Shankha ; Rubinfeld, Ronitt ; Yodpinyanee, Anak ( January 2020 , 11th Innovations in Theoretical Computer Science Conference, ITCS 2020)

   null (Ed.)

   Consider an algorithm performing a computation on a huge random object (for example a random graph or a "long" random walk). Is it necessary to generate the entire object prior to the computation, or is it possible to provide query access to the object and sample it incrementally "on-the-fly" (as requested by the algorithm)? Such an implementation should emulate the random object by answering queries in a manner consistent with an instance of the random object sampled from the true distribution (or close to it). This paradigm is useful when the algorithm is sub-linear and thus, sampling the entire object up front would ruin its efficiency. Our first set of results focus on undirected graphs with independent edge probabilities, i.e. each edge is chosen as an independent Bernoulli random variable. We provide a general implementation for this model under certain assumptions. Then, we use this to obtain the first efficient local implementations for the Erdös-Rényi G(n,p) model for all values of p, and the Stochastic Block model. As in previous local-access implementations for random graphs, we support Vertex-Pair and Next-Neighbor queries. In addition, we introduce a new Random-Neighbor query. Next, we give the first local-access implementation for All-Neighbors queries in the (sparse and directed) Kleinberg’s Small-World model. Our implementations require no pre-processing time, and answer each query using O(poly(log n)) time, random bits, and additional space. Next, we show how to implement random Catalan objects, specifically focusing on Dyck paths (balanced random walks on the integer line that are always non-negative). Here, we support Height queries to find the location of the walk, and First-Return queries to find the time when the walk returns to a specified location. This in turn can be used to implement Next-Neighbor queries on random rooted ordered trees, and Matching-Bracket queries on random well bracketed expressions (the Dyck language). Finally, we introduce two features to define a new model that: (1) allows multiple independent (and even simultaneous) instantiations of the same implementation, to be consistent with each other without the need for communication, (2) allows us to generate a richer class of random objects that do not have a succinct description. Specifically, we study uniformly random valid q-colorings of an input graph G with maximum degree Δ. This is in contrast to prior work in the area, where the relevant random objects are defined as a distribution with O(1) parameters (for example, n and p in the G(n,p) model). The distribution over valid colorings is instead specified via a "huge" input (the underlying graph G), that is far too large to be read by a sub-linear time algorithm. Instead, our implementation accesses G through local neighborhood probes, and is able to answer queries to the color of any given vertex in sub-linear time for q ≥ 9Δ, in a manner that is consistent with a specific random valid coloring of G. Furthermore, the implementation is memory-less, and can maintain consistency with non-communicating copies of itself. 

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