Christopher J. Rozell

Alec Helbling, Christopher J. Rozell, Matthew O'Shaughnessy et al.

Deep generative models have the capacity to render high fidelity images of content like human faces. Recently, there has been substantial progress in conditionally generating images with specific quantitative attributes, like the emotion conveyed by one's face. These methods typically require a user to explicitly quantify the desired intensity of a visual attribute. A limitation of this method is that many attributes, like how "angry" a human face looks, are difficult for a user to precisely quantify. However, a user would be able to reliably say which of two faces seems "angrier". Following this premise, we develop the $\textit{PrefGen}$ system, which allows users to control the relative attributes of generated images by presenting them with simple paired comparison queries of the form "do you prefer image $a$ or image $b$?" Using information from a sequence of query responses, we can estimate user preferences over a set of image attributes and perform preference-guided image editing and generation. Furthermore, to make preference localization feasible and efficient, we apply an active query selection strategy. We demonstrate the success of this approach using a StyleGAN2 generator on the task of human face editing. Additionally, we demonstrate how our approach can be combined with CLIP, allowing a user to edit the relative intensity of attributes specified by text prompts. Code at https://github.com/helblazer811/PrefGen.

Noga Mudrik, Yenho Chen, Eva Yezerets et al.

Learning interpretable representations of neural dynamics at a population level is a crucial first step to understanding how observed neural activity relates to perception and behavior. Models of neural dynamics often focus on either low-dimensional projections of neural activity, or on learning dynamical systems that explicitly relate to the neural state over time. We discuss how these two approaches are interrelated by considering dynamical systems as representative of flows on a low-dimensional manifold. Building on this concept, we propose a new decomposed dynamical system model that represents complex non-stationary and nonlinear dynamics of time series data as a sparse combination of simpler, more interpretable components. Our model is trained through a dictionary learning procedure, where we leverage recent results in tracking sparse vectors over time. The decomposed nature of the dynamics is more expressive than previous switched approaches for a given number of parameters and enables modeling of overlapping and non-stationary dynamics. In both continuous-time and discrete-time instructional examples we demonstrate that our model can well approximate the original system, learn efficient representations, and capture smooth transitions between dynamical modes, focusing on intuitive low-dimensional non-stationary linear and nonlinear systems. Furthermore, we highlight our model's ability to efficiently capture and demix population dynamics generated from multiple independent subnetworks, a task that is computationally impractical for switched models. Finally, we apply our model to neural "full brain" recordings of C. elegans data, illustrating a diversity of dynamics that is obscured when classified into discrete states.

Han Lun Yap, Christopher J. Rozell

Takens' Embedding Theorem remarkably established that concatenating M previous outputs of a dynamical system into a vector (called a delay coordinate map) can be a one-to-one mapping of a low-dimensional attractor from the system state space. However, Takens' theorem is fragile in the sense that even small imperfections can induce arbitrarily large errors in this attractor representation. We extend Takens' result to establish deterministic, explicit and non-asymptotic sufficient conditions for a delay coordinate map to form a stable embedding in the restricted case of linear dynamical systems and observation functions. Our work is inspired by the field of Compressive Sensing (CS), where results guarantee that low-dimensional signal families can be robustly reconstructed if they are stably embedded by a measurement operator. However, in contrast to typical CS results, i) our sufficient conditions are independent of the size of the ambient state space, and ii) some system and measurement pairs have fundamental limits on the conditioning of the embedding (i.e., how close it is to an isometry), meaning that further measurements beyond some point add no further significant value. We use several simple simulations to explore the conditions of the main results, including the tightness of the bounds and the convergence speed of the stable embedding. We also present an example task of estimating the attractor dimension from time-series data to highlight the value of stable embeddings over traditional Takens' embeddings.

Marissa Connor, Bruno Olshausen, Christopher Rozell

When interacting in a three dimensional world, humans must estimate 3D structure from visual inputs projected down to two dimensional retinal images. It has been shown that humans use the persistence of object shape over motion-induced transformations as a cue to resolve depth ambiguity when solving this underconstrained problem. With the aim of understanding how biological vision systems may internally represent 3D transformations, we propose a computational model, based on a generative manifold model, which can be used to infer 3D structure from the motion of 2D points. Our model can also learn representations of the transformations with minimal supervision, providing a proof of concept for how humans may develop internal representations on a developmental or evolutionary time scale. Focused on rotational motion, we show how our model infers depth from moving 2D projected points, learns 3D rotational transformations from 2D training stimuli, and compares to human performance on psychophysical structure-from-motion experiments.

Kion Fallah, Christopher J. Rozell

Sparse coding strategies have been lauded for their parsimonious representations of data that leverage low dimensional structure. However, inference of these codes typically relies on an optimization procedure with poor computational scaling in high-dimensional problems. For example, sparse inference in the representations learned in the high-dimensional intermediary layers of deep neural networks (DNNs) requires an iterative minimization to be performed at each training step. As such, recent, quick methods in variational inference have been proposed to infer sparse codes by learning a distribution over the codes with a DNN. In this work, we propose a new approach to variational sparse coding that allows us to learn sparse distributions by thresholding samples, avoiding the use of problematic relaxations. We first evaluate and analyze our method by training a linear generator, showing that it has superior performance, statistical efficiency, and gradient estimation compared to other sparse distributions. We then compare to a standard variational autoencoder using a DNN generator on the Fashion MNIST and CelebA datasets

Yenho Chen, Noga Mudrik, Kyle A. Johnsen et al.

Time-varying linear state-space models are powerful tools for obtaining mathematically interpretable representations of neural signals. For example, switching and decomposed models describe complex systems using latent variables that evolve according to simple locally linear dynamics. However, existing methods for latent variable estimation are not robust to dynamical noise and system nonlinearity due to noise-sensitive inference procedures and limited model formulations. This can lead to inconsistent results on signals with similar dynamics, limiting the model's ability to provide scientific insight. In this work, we address these limitations and propose a probabilistic approach to latent variable estimation in decomposed models that improves robustness against dynamical noise. Additionally, we introduce an extended latent dynamics model to improve robustness against system nonlinearities. We evaluate our approach on several synthetic dynamical systems, including an empirically-derived brain-computer interface experiment, and demonstrate more accurate latent variable inference in nonlinear systems with diverse noise conditions. Furthermore, we apply our method to a real-world clinical neurophysiology dataset, illustrating the ability to identify interpretable and coherent structure where previous models cannot.

Christoffer Loeffler, Kion Fallah, Stefano Fenu et al.

Humans innately measure distance between instances in an unlabeled dataset using an unknown similarity function. Distance metrics can only serve as proxy for similarity in information retrieval of similar instances. Learning a good similarity function from human annotations improves the quality of retrievals. This work uses deep metric learning to learn these user-defined similarity functions from few annotations for a large football trajectory dataset. We adapt an entropy-based active learning method with recent work from triplet mining to collect easy-to-answer but still informative annotations from human participants and use them to train a deep convolutional network that generalizes to unseen samples. Our user study shows that our approach improves the quality of the information retrieval compared to a previous deep metric learning approach that relies on a Siamese network. Specifically, we shed light on the strengths and weaknesses of passive sampling heuristics and active learners alike by analyzing the participants' response efficacy. To this end, we collect accuracy, algorithmic time complexity, the participants' fatigue and time-to-response, qualitative self-assessment and statements, as well as the effects of mixed-expertise annotators and their consistency on model performance and transfer-learning.

Kion Fallah, Alec Helbling, Kyle A. Johnsen et al.

Self-supervised learning of deep neural networks has become a prevalent paradigm for learning representations that transfer to a variety of downstream tasks. Similar to proposed models of the ventral stream of biological vision, it is observed that these networks lead to a separation of category manifolds in the representations of the penultimate layer. Although this observation matches the manifold hypothesis of representation learning, current self-supervised approaches are limited in their ability to explicitly model this manifold. Indeed, current approaches often only apply augmentations from a pre-specified set of "positive pairs" during learning. In this work, we propose a contrastive learning approach that directly models the latent manifold using Lie group operators parameterized by coefficients with a sparsity-promoting prior. A variational distribution over these coefficients provides a generative model of the manifold, with samples which provide feature augmentations applicable both during contrastive training and downstream tasks. Additionally, learned coefficient distributions provide a quantification of which transformations are most likely at each point on the manifold while preserving identity. We demonstrate benefits in self-supervised benchmarks for image datasets, as well as a downstream semi-supervised task. In the former case, we demonstrate that the proposed methods can effectively apply manifold feature augmentations and improve learning both with and without a projection head. In the latter case, we demonstrate that feature augmentations sampled from learned Lie group operators can improve classification performance when using few labels.

Alec Helbling, Christopher John Rozell, Matthew O'Shaughnessy et al.

Isolating and controlling specific features in the outputs of generative models in a user-friendly way is a difficult and open-ended problem. We develop techniques that allow an oracle user to generate an image they are envisioning in their head by answering a sequence of relative queries of the form \textit{"do you prefer image $a$ or image $b$?"} Our framework consists of a Conditional VAE that uses the collected relative queries to partition the latent space into preference-relevant features and non-preference-relevant features. We then use the user's responses to relative queries to determine the preference-relevant features that correspond to their envisioned output image. Additionally, we develop techniques for modeling the uncertainty in images' predicted preference-relevant features, allowing our framework to generalize to scenarios in which the relative query training set contains noise.

Belén Martín-Urcelay, Yoonsang Lee, Matthieu R. Bloch et al.

Integrating human expertise into machine learning systems often reduces the role of experts to labeling oracles, a paradigm that limits the amount of information exchanged and fails to capture the nuances of human judgment. We address this challenge by developing a human-in-the-loop framework to learn binary classifiers with rich query types, consisting of item ranking and exemplar selection. We first introduce probabilistic human response models for these rich queries motivated by the relationship experimentally observed between the perceived implicit score of an item and its distance to the unknown classifier. Using these models, we then design active learning algorithms that leverage the rich queries to increase the information gained per interaction. We provide theoretical bounds on sample complexity and develop a tractable and computationally efficient variational approximation. Through experiments with simulated annotators derived from crowdsourced word-sentiment and image-aesthetic datasets, we demonstrate significant reductions on sample complexity. We further extend active learning strategies to select queries that maximize information rate, explicitly balancing informational value against annotation cost. This algorithm in the word sentiment classification task reduces learning time by more than 57\% compared to traditional label-only active learning.

Vivek Anand, Alec Helbling, Mark Davenport et al.

Learning the intrinsic dimensionality of subjective perceptual spaces such as taste, smell, or aesthetics from ordinal data is a challenging problem. We introduce LORE (Low Rank Ordinal Embedding), a scalable framework that jointly learns both the intrinsic dimensionality and an ordinal embedding from noisy triplet comparisons of the form, "Is A more similar to B than C?". Unlike existing methods that require the embedding dimension to be set apriori, LORE regularizes the solution using the nonconvex Schatten-$p$ quasi norm, enabling automatic joint recovery of both the ordinal embedding and its dimensionality. We optimize this joint objective via an iteratively reweighted algorithm and establish convergence guarantees. Extensive experiments on synthetic datasets, simulated perceptual spaces, and real world crowdsourced ordinal judgements show that LORE learns compact, interpretable and highly accurate low dimensional embeddings that recover the latent geometry of subjective percepts. By simultaneously inferring both the intrinsic dimensionality and ordinal embeddings, LORE enables more interpretable and data efficient perceptual modeling in psychophysics and opens new directions for scalable discovery of low dimensional structure from ordinal data in machine learning.

Noga Mudrik, Eva Yezerets, Yenho Chen et al.

Identifying latent interactions within complex systems is key to unlocking deeper insights into their operational dynamics, including how their elements affect each other and contribute to the overall system behavior. For instance, in neuroscience, discovering neuron-to-neuron interactions is essential for understanding brain function; in ecology, recognizing the interactions among populations is key for understanding complex ecosystems. Such systems, often modeled as dynamical systems, typically exhibit noisy high-dimensional and non-stationary temporal behavior that renders their identification challenging. Existing dynamical system identification methods often yield operators that accurately capture short-term behavior but fail to predict long-term trends, suggesting an incomplete capture of the underlying process. Methods that consider extended forecasts (e.g., recurrent neural networks) lack explicit representations of element-wise interactions and require substantial training data, thereby failing to capture interpretable network operators. Here we introduce Lookahead-driven Inference of Networked Operators for Continuous Stability (LINOCS), a robust learning procedure for identifying hidden dynamical interactions in noisy time-series data. LINOCS integrates several multi-step predictions with adaptive weights during training to recover dynamical operators that can yield accurate long-term predictions. We demonstrate LINOCS' ability to recover the ground truth dynamical operators underlying synthetic time-series data for multiple dynamical systems models (including linear, piece-wise linear, time-changing linear systems' decomposition, and regularized linear time-varying systems) as well as its capability to produce meaningful operators with robust reconstructions through various real-world examples.

Yenho Chen, Maxwell A. Xu, James M. Rehg et al.

The effectiveness of self-supervised learning (SSL) for physiological time series depends on the ability of a pretraining objective to preserve information about the underlying physiological state while filtering out unrelated noise. However, existing strategies are limited due to reliance on heuristic principles or poorly constrained generative tasks. To address this limitation, we propose a pretraining framework that exploits the information structure of a dynamical systems generative model across multiple time-series. This framework reveals our key insight that class identity can be efficiently captured by extracting information about the generative variables related to the system parameters shared across similar time series samples, while noise unique to individual samples should be discarded. Building on this insight, we propose PULSE, a cross-reconstruction-based pretraining objective for physiological time series datasets that explicitly extracts system information while discarding non-transferrable sample-specific ones. We establish theory that provides sufficient conditions for the system information to be recovered, and empirically validate it using a synthetic dynamical systems experiment. Furthermore, we apply our method to diverse real-world datasets, demonstrating that PULSE learns representations that can broadly distinguish semantic classes, increase label efficiency, and improve transfer learning.

Marissa Connor, Kion Fallah, Christopher Rozell

Many machine learning techniques incorporate identity-preserving transformations into their models to generalize their performance to previously unseen data. These transformations are typically selected from a set of functions that are known to maintain the identity of an input when applied (e.g., rotation, translation, flipping, and scaling). However, there are many natural variations that cannot be labeled for supervision or defined through examination of the data. As suggested by the manifold hypothesis, many of these natural variations live on or near a low-dimensional, nonlinear manifold. Several techniques represent manifold variations through a set of learned Lie group operators that define directions of motion on the manifold. However, these approaches are limited because they require transformation labels when training their models and they lack a method for determining which regions of the manifold are appropriate for applying each specific operator. We address these limitations by introducing a learning strategy that does not require transformation labels and developing a method that learns the local regions where each operator is likely to be used while preserving the identity of inputs. Experiments on MNIST and Fashion MNIST highlight our model's ability to learn identity-preserving transformations on multi-class datasets. Additionally, we train on CelebA to showcase our model's ability to learn semantically meaningful transformations on complex datasets in an unsupervised manner.

Ayse S. Cakmak, Samuel Densen, Gabriel Najarro et al.

Objective: Worldwide, heart failure (HF) is a major cause of morbidity and mortality and one of the leading causes of hospitalization. Early detection of HF symptoms and pro-active management may reduce adverse events. Approach: Twenty-eight participants were monitored using a smartphone app after discharge from hospitals, and each clinical event during the enrollment (N=110 clinical events) was recorded. Motion, social, location, and clinical survey data collected via the smartphone-based monitoring system were used to develop and validate an algorithm for predicting or classifying HF decompensation events (hospitalizations or clinic visit) versus clinic monitoring visits in which they were determined to be compensated or stable. Models based on single modality as well as early and late fusion approaches combining patient-reported outcomes and passive smartphone data were evaluated. Results: The highest AUCPr for classifying decompensation with a late fusion approach was 0.80 using leave one subject out cross-validation. Significance: Passively collected data from smartphones, especially when combined with weekly patient-reported outcomes, may reflect behavioral and physiological changes due to HF and thus could enable prediction of HF decompensation.

Gregory Canal, Matthieu Bloch, Christopher Rozell

The iterative selection of examples for labeling in active machine learning is conceptually similar to feedback channel coding in information theory: in both tasks, the objective is to seek a minimal sequence of actions to encode information in the presence of noise. While this high-level overlap has been previously noted, there remain open questions on how to best formulate active learning as a communications system to leverage existing analysis and algorithms in feedback coding. In this work, we formally identify and leverage the structural commonalities between the two problems, including the characterization of encoder and noisy channel components, to design a new algorithm. Specifically, we develop an optimal transport-based feedback coding scheme called Approximate Posterior Matching (APM) for the task of active example selection and explore its application to Bayesian logistic regression, a popular model in active learning. We evaluate APM on a variety of datasets and demonstrate learning performance comparable to existing active learning methods, at a reduced computational cost. These results demonstrate the potential of directly deploying concepts from feedback channel coding to design efficient active learning strategies.

Matthew O'Shaughnessy, Gregory Canal, Marissa Connor et al.

We develop a method for generating causal post-hoc explanations of black-box classifiers based on a learned low-dimensional representation of the data. The explanation is causal in the sense that changing learned latent factors produces a change in the classifier output statistics. To construct these explanations, we design a learning framework that leverages a generative model and information-theoretic measures of causal influence. Our objective function encourages both the generative model to faithfully represent the data distribution and the latent factors to have a large causal influence on the classifier output. Our method learns both global and local explanations, is compatible with any classifier that admits class probabilities and a gradient, and does not require labeled attributes or knowledge of causal structure. Using carefully controlled test cases, we provide intuition that illuminates the function of our objective. We then demonstrate the practical utility of our method on image recognition tasks.

Marissa C. Connor, Gregory H. Canal, Christopher J. Rozell

The manifold hypothesis states that high-dimensional data can be modeled as lying on or near a low-dimensional, nonlinear manifold. Variational Autoencoders (VAEs) approximate this manifold by learning mappings from low-dimensional latent vectors to high-dimensional data while encouraging a global structure in the latent space through the use of a specified prior distribution. When this prior does not match the structure of the true data manifold, it can lead to a less accurate model of the data. To resolve this mismatch, we introduce the Variational Autoencoder with Learned Latent Structure (VAELLS) which incorporates a learnable manifold model into the latent space of a VAE. This enables us to learn the nonlinear manifold structure from the data and use that structure to define a prior in the latent space. The integration of a latent manifold model not only ensures that our prior is well-matched to the data, but also allows us to define generative transformation paths in the latent space and describe class manifolds with transformations stemming from examples of each class. We validate our model on examples with known latent structure and also demonstrate its capabilities on a real-world dataset.

Marissa Connor, Christopher Rozell

Deep generative networks have been widely used for learning mappings from a low-dimensional latent space to a high-dimensional data space. In many cases, data transformations are defined by linear paths in this latent space. However, the Euclidean structure of the latent space may be a poor match for the underlying latent structure in the data. In this work, we incorporate a generative manifold model into the latent space of an autoencoder in order to learn the low-dimensional manifold structure from the data and adapt the latent space to accommodate this structure. In particular, we focus on applications in which the data has closed transformation paths which extend from a starting point and return to nearly the same point. Through experiments on data with natural closed transformation paths, we show that this model introduces the ability to learn the latent dynamics of complex systems, generate transformation paths, and classify samples that belong on the same transformation path.

Gregory Canal, Stefano Fenu, Christopher Rozell

Many machine learning tasks such as clustering, classification, and dataset search benefit from embedding data points in a space where distances reflect notions of relative similarity as perceived by humans. A common way to construct such an embedding is to request triplet similarity queries to an oracle, comparing two objects with respect to a reference. This work generalizes triplet queries to tuple queries of arbitrary size that ask an oracle to rank multiple objects against a reference, and introduces an efficient and robust adaptive selection method called InfoTuple that uses a novel approach to mutual information maximization. We show that the performance of InfoTuple at various tuple sizes exceeds that of the state-of-the-art adaptive triplet selection method on synthetic tests and new human response datasets, and empirically demonstrate the significant gains in efficiency and query consistency achieved by querying larger tuples instead of triplets.

John Lee, Max Dabagia, Eva L. Dyer et al.

In many machine learning applications, it is necessary to meaningfully aggregate, through alignment, different but related datasets. Optimal transport (OT)-based approaches pose alignment as a divergence minimization problem: the aim is to transform a source dataset to match a target dataset using the Wasserstein distance as a divergence measure. We introduce a hierarchical formulation of OT which leverages clustered structure in data to improve alignment in noisy, ambiguous, or multimodal settings. To solve this numerically, we propose a distributed ADMM algorithm that also exploits the Sinkhorn distance, thus it has an efficient computational complexity that scales quadratically with the size of the largest cluster. When the transformation between two datasets is unitary, we provide performance guarantees that describe when and how well aligned cluster correspondences can be recovered with our formulation, as well as provide worst-case dataset geometry for such a strategy. We apply this method to synthetic datasets that model data as mixtures of low-rank Gaussians and study the impact that different geometric properties of the data have on alignment. Next, we applied our approach to a neural decoding application where the goal is to predict movement directions and instantaneous velocities from populations of neurons in the macaque primary motor cortex. Our results demonstrate that when clustered structure exists in datasets, and is consistent across trials or time points, a hierarchical alignment strategy that leverages such structure can provide significant improvements in cross-domain alignment.

Gregory H. Canal, Andrew K. Massimino, Mark A. Davenport et al.

Suppose that we wish to estimate a user's preference vector $w$ from paired comparisons of the form "does user $w$ prefer item $p$ or item $q$?," where both the user and items are embedded in a low-dimensional Euclidean space with distances that reflect user and item similarities. Such observations arise in numerous settings, including psychometrics and psychology experiments, search tasks, advertising, and recommender systems. In such tasks, queries can be extremely costly and subject to varying levels of response noise; thus, we aim to actively choose pairs that are most informative given the results of previous comparisons. We provide new theoretical insights into the benefits and challenges of greedy information maximization in this setting, and develop two novel strategies that maximize lower bounds on information gain and are simpler to analyze and compute respectively. We use simulated responses from a real-world dataset to validate our strategies through their similar performance to greedy information maximization, and their superior preference estimation over state-of-the-art selection methods as well as random queries.

Adam Charles, Dong Yin, Christopher Rozell

Recurrent neural networks (RNNs) have drawn interest from machine learning researchers because of their effectiveness at preserving past inputs for time-varying data processing tasks. To understand the success and limitations of RNNs, it is critical that we advance our analysis of their fundamental memory properties. We focus on echo state networks (ESNs), which are RNNs with simple memoryless nodes and random connectivity. In most existing analyses, the short-term memory (STM) capacity results conclude that the ESN network size must scale linearly with the input size for unstructured inputs. The main contribution of this paper is to provide general results characterizing the STM capacity for linear ESNs with multidimensional input streams when the inputs have common low-dimensional structure: sparsity in a basis or significant statistical dependence between inputs. In both cases, we show that the number of nodes in the network must scale linearly with the information rate and poly-logarithmically with the ambient input dimension. The analysis relies on advanced applications of random matrix theory and results in explicit non-asymptotic bounds on the recovery error. Taken together, this analysis provides a significant step forward in our understanding of the STM properties in RNNs.

Adam Charles, Aurele Balavoine, Christopher Rozell

Despite the importance of sparsity signal models and the increasing prevalence of high-dimensional streaming data, there are relatively few algorithms for dynamic filtering of time-varying sparse signals. Of the existing algorithms, fewer still provide strong performance guarantees. This paper examines two algorithms for dynamic filtering of sparse signals that are based on efficient l1 optimization methods. We first present an analysis for one simple algorithm (BPDN-DF) that works well when the system dynamics are known exactly. We then introduce a novel second algorithm (RWL1-DF) that is more computationally complex than BPDN-DF but performs better in practice, especially in the case where the system dynamics model is inaccurate. Robustness to model inaccuracy is achieved by using a hierarchical probabilistic data model and propagating higher-order statistics from the previous estimate (akin to Kalman filtering) in the sparse inference process. We demonstrate the properties of these algorithms on both simulated data as well as natural video sequences. Taken together, the algorithms presented in this paper represent the first strong performance analysis of dynamic filtering algorithms for time-varying sparse signals as well as state-of-the-art performance in this emerging application.

Adam S. Charles, Han Lun Yap, Christopher J. Rozell

Cortical networks are hypothesized to rely on transient network activity to support short term memory (STM). In this paper we study the capacity of randomly connected recurrent linear networks for performing STM when the input signals are approximately sparse in some basis. We leverage results from compressed sensing to provide rigorous non asymptotic recovery guarantees, quantifying the impact of the input sparsity level, the input sparsity basis, and the network characteristics on the system capacity. Our analysis demonstrates that network memory capacities can scale superlinearly with the number of nodes, and in some situations can achieve STM capacities that are much larger than the network size. We provide perfect recovery guarantees for finite sequences and recovery bounds for infinite sequences. The latter analysis predicts that network STM systems may have an optimal recovery length that balances errors due to omission and recall mistakes. Furthermore, we show that the conditions yielding optimal STM capacity can be embodied in several network topologies, including networks with sparse or dense connectivities.