arXiv subjects

Peter Allen

Publications and source records attributed to Peter Allen.

At least 37 records · Page 2Linked to original sources

arXiv2022

Partitioning a 2-edge-coloured graph of minimum degree $2n/3 + o(n)$ into three monochromatic cycles

Lehel conjectured in the 1970s that every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomassé. However, the host graph $G$ does not have to be complete. It it suffices to require that $G$ has minimum degree at least $3n/4$, where $n$ is the order of $G$, as was shown recently by Letzter, confirming a conjecture of Balogh, Barát, Gerbner, Gyárfás and Sárközy. This degree condition is asymptotically tight. Here we continue this line of research, by proving that for every red and blue edge-colouring of an $n$-vertex graph of minimum degree at least $2n/3 + o(n)$, there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight.

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arXiv2021

Resilience for tight Hamiltonicity

We prove that random hypergraphs are asymptotically almost surely resiliently Hamiltonian. Specifically, for any $γ>0$ and $k\ge3$, we show that asymptotically almost surely, every subgraph of the binomial random $k$-uniform hypergraph $G^{(k)}\big(n,n^{γ-1}\big)$ in which all $(k-1)$-sets are contained in at least $\big(\tfrac12+2γ\big)pn$ edges has a tight Hamilton cycle. This is a cyclic ordering of the $n$ vertices such that each consecutive $k$ vertices forms an edge.

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arXiv2021

CLAMGen: Closed-Loop Arm Motion Generation via Multi-view Vision-Based RL

We propose a vision-based reinforcement learning (RL) approach for closed-loop trajectory generation in an arm reaching problem. Arm trajectory generation is a fundamental robotics problem which entails finding collision-free paths to move the robot's body (e.g. arm) in order to satisfy a goal (e.g. place end-effector at a point). While classical methods typically require the model of the environment to solve a planning, search or optimization problem, learning-based approaches hold the promise of directly mapping from observations to robot actions. However, learning a collision-avoidance policy using RL remains a challenge for various reasons, including, but not limited to, partial observability, poor exploration, low sample efficiency, and learning instabilities. To address these challenges, we present a residual-RL method that leverages a greedy goal-reaching RL policy as the base to improve exploration, and the base policy is augmented with residual state-action values and residual actions learned from images to avoid obstacles. Further more, we introduce novel learning objectives and techniques to improve 3D understanding from multiple image views and sample efficiency of our algorithm. Compared to RL baselines, our method achieves superior performance in terms of success rate.

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arXiv2020

The Ramsey number of a long even cycle versus a star

We find the exact value of the Ramsey number $R(C_{2\ell},K_{1,n})$, when $\ell$ and $n=O(\ell^{10/9})$ are large. Our result is closely related to the behaviour of Turán number $ex(N, C_{2\ell})$ for an even cycle whose length grows quickly with $N$.

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arXiv2020

Learning Your Way Without Map or Compass: Panoramic Target Driven Visual Navigation

We present a robot navigation system that uses an imitation learning framework to successfully navigate in complex environments. Our framework takes a pre-built 3D scan of a real environment and trains an agent from pre-generated expert trajectories to navigate to any position given a panoramic view of the goal and the current visual input without relying on map, compass, odometry, or relative position of the target at runtime. Our end-to-end trained agent uses RGB and depth (RGBD) information and can handle large environments (up to $1031m^2$) across multiple rooms (up to $40$) and generalizes to unseen targets. We show that when compared to several baselines our method (1) requires fewer training examples and less training time, (2) reaches the goal location with higher accuracy, and (3) produces better solutions with shorter paths for long-range navigation tasks.

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arXiv2020

Accelerated Robot Learning via Human Brain Signals

In reinforcement learning (RL), sparse rewards are a natural way to specify the task to be learned. However, most RL algorithms struggle to learn in this setting since the learning signal is mostly zeros. In contrast, humans are good at assessing and predicting the future consequences of actions and can serve as good reward/policy shapers to accelerate the robot learning process. Previous works have shown that the human brain generates an error-related signal, measurable using electroencephelography (EEG), when the human perceives the task being done erroneously. In this work, we propose a method that uses evaluative feedback obtained from human brain signals measured via scalp EEG to accelerate RL for robotic agents in sparse reward settings. As the robot learns the task, the EEG of a human observer watching the robot attempts is recorded and decoded into noisy error feedback signal. From this feedback, we use supervised learning to obtain a policy that subsequently augments the behavior policy and guides exploration in the early stages of RL. This bootstraps the RL learning process to enable learning from sparse reward. Using a robotic navigation task as a test bed, we show that our method achieves a stable obstacle-avoidance policy with high success rate, outperforming learning from sparse rewards only that struggles to achieve obstacle avoidance behavior or fails to advance to the goal.

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arXiv2020

Maximizing BCI Human Feedback using Active Learning

Recent advancements in \textit{Learning from Human Feedback} present an effective way to train robot agents via inputs from non-expert humans, without a need for a specially designed reward function. However, this approach needs a human to be present and attentive during robot learning to provide evaluative feedback. In addition, the amount of feedback needed grows with the level of task difficulty and the quality of human feedback might decrease over time because of fatigue. To overcome these limitations and enable learning more robot tasks with higher complexities, there is a need to maximize the quality of expensive feedback received and reduce the amount of human cognitive involvement required. In this work, we present an approach that uses active learning to smartly choose queries for the human supervisor based on the uncertainty of the robot and effectively reduces the amount of feedback needed to learn a given task. We also use a novel multiple buffer system to improve robustness to feedback noise and guard against catastrophic forgetting as the robot learning evolves. This makes it possible to learn tasks with more complexity using lesser amounts of human feedback compared to previous methods. We demonstrate the utility of our proposed method on a robot arm reaching task where the robot learns to reach a location in 3D without colliding with obstacles. Our approach is able to learn this task faster, with less human feedback and cognitive involvement, compared to previous methods that do not use active learning.

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arXiv2020

The Bandwidth Theorem in Sparse Graphs

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. We provide sparse analogues of this statement in random graphs as well as pseudorandom graphs. More precisely, we show that for $p\gg \big(\tfrac{\log n}{n}\big)^{1/Δ}$ asymptotically almost surely each spanning subgraph $G$ of $G(n,p)$ with minimum degree $\big(\tfrac{k-1}{k}+o(1)\big)pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $Δ$, bandwidth $o(n)$, and at least $C p^{-2}$ vertices not contained in any triangle. A similar result is shown for sufficiently bijumbled graphs, which, to the best of our knowledge, is the first resilience result in pseudorandom graphs for a rich class of spanning subgraphs. Finally, we provide improved results for $H$ with small degeneracy, which in particular imply a resilience result in $G(n,p)$ with respect to the containment of spanning bounded degree trees for $p\gg \big(\tfrac{\log n}{n}\big)^{1/3}$.

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arXiv2019

A spanning bandwidth theorem in random graphs

The bandwidth theorem [Mathematische Annalen, 343(1):175--205, 2009] states that any $n$-vertex graph $G$ with minimum degree $(\frac{k-1}{k}+o(1))n$ contains all $n$-vertex $k$-colourable graphs $H$ with bounded maximum degree and bandwidth $o(n)$. In [arXiv:1612.00661] a random graph analogue of this statement is proved: for $p\gg (\frac{\log n}{n})^{1/Δ}$ a.a.s. each spanning subgraph $G$ of $G(n,p)$ with minimum degree $(\frac{k-1}{k}+o(1))pn$ contains all $n$-vertex $k$-colourable graphs $H$ with maximum degree $Δ$, bandwidth $o(n)$, and at least $C p^{-2}$ vertices not contained in any triangle. This restriction on vertices in triangles is necessary, but limiting. In this paper we consider how it can be avoided. A special case of our main result is that, under the same conditions, if additionally all vertex neighbourhoods in $G$ contain many copies of $K_Δ$ then we can drop the restriction on $H$ that $Cp^{-2}$ vertices should not be in triangles.

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arXiv2019

MAT: Multi-Fingered Adaptive Tactile Grasping via Deep Reinforcement Learning

Vision-based grasping systems typically adopt an open-loop execution of a planned grasp. This policy can fail due to many reasons, including ubiquitous calibration error. Recovery from a failed grasp is further complicated by visual occlusion, as the hand is usually occluding the vision sensor as it attempts another open-loop regrasp. This work presents MAT, a tactile closed-loop method capable of realizing grasps provided by a coarse initial positioning of the hand above an object. Our algorithm is a deep reinforcement learning (RL) policy optimized through the clipped surrogate objective within a maximum entropy RL framework to balance exploitation and exploration. The method utilizes tactile and proprioceptive information to act through both fine finger motions and larger regrasp movements to execute stable grasps. A novel curriculum of action motion magnitude makes learning more tractable and helps turn common failure cases into successes. Careful selection of features that exhibit small sim-to-real gaps enables this tactile grasping policy, trained purely in simulation, to transfer well to real world environments without the need for additional learning. Experimentally, this methodology improves over a vision-only grasp success rate substantially on a multi-fingered robot hand. When this methodology is used to realize grasps from coarse initial positions provided by a vision-only planner, the system is made dramatically more robust to calibration errors in the camera-robot transform.

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arXiv2019

Almost all trees are almost graceful

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each c>0 and for all n>n_0(c). Suppose that (i) the maximum degree of T is bounded by O(n/log n), and (ii) the vertex labels are chosen from the set {1,2,..., (1+c)n}. Then there is an injective labelling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labelling. As a consequence, for any such tree T we can pack (2+2c)n-1 copies of T into the complete graph of order (2+2c)n-1 cyclically. This proves an approximate version of the Ringel-Kotzig conjecture (which asserts the existence of a cyclic packing of 2n-1 copies of any T into the complete graph of order 2n-1) for these trees. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labelling with high probability.

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arXiv2019

Perfectly packing graphs with bounded degeneracy and many leaves

We prove that one can perfectly pack degenerate graphs into complete or dense $n$-vertex quasirandom graphs, provided that all the degenerate graphs have maximum degree $o(\frac{n}{\log n})$, and in addition $Ω(n)$ of them have at most $(1-Ω(1))n$ vertices and $Ω(n)$ leaves. This proves Ringel's conjecture and the Gyárfás Tree Packing Conjecture for all but an exponentially small fraction of trees (or sequences of trees, respectively).

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arXiv2019

Multi-Modal Geometric Learning for Grasping and Manipulation

This work provides an architecture that incorporates depth and tactile information to create rich and accurate 3D models useful for robotic manipulation tasks. This is accomplished through the use of a 3D convolutional neural network (CNN). Offline, the network is provided with both depth and tactile information and trained to predict the object's geometry, thus filling in regions of occlusion. At runtime, the network is provided a partial view of an object. Tactile information is acquired to augment the captured depth information. The network can then reason about the object's geometry by utilizing both the collected tactile and depth information. We demonstrate that even small amounts of additional tactile information can be incredibly helpful in reasoning about object geometry. This is particularly true when information from depth alone fails to produce an accurate geometric prediction. Our method is benchmarked against and outperforms other visual-tactile approaches to general geometric reasoning. We also provide experimental results comparing grasping success with our method.

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arXiv2019

Regularity inheritance in pseudorandom graphs

Advancing the sparse regularity method, we prove one-sided and two-sided regularity inheritance lemmas for subgraphs of bijumbled graphs, improving on results of Conlon, Fox and Zhao [Adv. Math. 256 (2014), 206--290]. These inheritance lemmas also imply improved $H$-counting lemmas for subgraphs of bijumbled graphs, for some $H$.

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arXiv2019

Regularity inheritance in hypergraphs

We give a new approach to handling hypergraph regularity. This approach allows for vertex-by-vertex embedding into regular partitions of hypergraphs, and generalises to regular partitions of sparse hypergraphs. We also prove a corresponding sparse hypergraph regularity lemma.

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arXiv2018

Human Robot Interface for Assistive Grasping

This work describes a new human-in-the-loop (HitL) assistive grasping system for individuals with varying levels of physical capabilities. We investigated the feasibility of using four potential input devices with our assistive grasping system interface, using able-bodied individuals to define a set of quantitative metrics that could be used to assess an assistive grasping system. We then took these measurements and created a generalized benchmark for evaluating the effectiveness of any arbitrary input device into a HitL grasping system. The four input devices were a mouse, a speech recognition device, an assistive switch, and a novel sEMG device developed by our group that was connected either to the forearm or behind the ear of the subject. These preliminary results provide insight into how different interface devices perform for generalized assistive grasping tasks and also highlight the potential of sEMG based control for severely disabled individuals.

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arXiv2017

Finding tight Hamilton cycles in random hypergraphs faster

In an $r$-uniform hypergraph on $n$ vertices a tight Hamilton cycle consists of $n$ edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of $r$ vertices. We provide a first deterministic polynomial time algorithm, which finds a.a.s. tight Hamilton cycles in random $r$-uniform hypergraphs with edge probability at least $C \log^3n/n$. Our result partially answers a question of Dudek and Frieze [Random Structures & Algorithms 42 (2013), 374-385] who proved that tight Hamilton cycles exists already for $p=ω(1/n)$ for $r=3$ and $p=(e + o(1))/n$ for $r\ge 4$ using a second moment argument. Moreover our algorithm is superior to previous results of Allen, Böttcher, Kohayakawa and Person [Random Structures & Algorithms 46 (2015), 446-465] and Nenadov and Škorić [arXiv:1601.04034] in various ways: the algorithm of Allen et al. is a randomised polynomial time algorithm working for edge probabilities $p\ge n^{-1+\varepsilon}$, while the algorithm of Nenadov and Škorić is a randomised quasipolynomial time algorithm working for edge probabilities $p\ge C\log^8n/n$.

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arXiv2017

Shape Completion Enabled Robotic Grasping

This work provides an architecture to enable robotic grasp planning via shape completion. Shape completion is accomplished through the use of a 3D convolutional neural network (CNN). The network is trained on our own new open source dataset of over 440,000 3D exemplars captured from varying viewpoints. At runtime, a 2.5D pointcloud captured from a single point of view is fed into the CNN, which fills in the occluded regions of the scene, allowing grasps to be planned and executed on the completed object. Runtime shape completion is very rapid because most of the computational costs of shape completion are borne during offline training. We explore how the quality of completions vary based on several factors. These include whether or not the object being completed existed in the training data and how many object models were used to train the network. We also look at the ability of the network to generalize to novel objects allowing the system to complete previously unseen objects at runtime. Finally, experimentation is done both in simulation and on actual robotic hardware to explore the relationship between completion quality and the utility of the completed mesh model for grasping.

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