Pengcheng You, a Postdoctoral researcher from our lab, will join Peking University as an Assistant Professor at College of Engineering in Spring 2022! Congratulations!

# News

## Hancheng passes his GBO

Hancheng Min, ECE Ph.D. student from our lab, passed his Graduate Board Oral Examination! Congratulations!

## Agustin joins NetDLab

Agustin Castellano joins our lab as a new Ph.D. student at ECE! Welcome!

## Tianqi passes his GBO

Tianqi Zheng, ECE Ph.D. student from our lab, passed his Graduate Board Oral Examination! Congratulations!

## 1 paper accepted to CDC 21

Our paper on tight inner approximation of the positive-semidefinite cone [1] have been accepted to IEEE Conference on Decision and Control!

[Bibtex] [Abstract] [Download PDF]

We investigate the problem of finding tight inner approximations of large dimensional positive semidefinite (PSD) cones. To solve this problem, we develop a novel decomposition framework of the PSD cone by means of conical combinations of smaller dimensional sub-cones. We show that many inner approximation techniques could be summarized within this framework, including the set of (scaled) diagonally dominant matrices, Factor-width k matrices, and Chordal Sparse matrices. Furthermore, we provide a more flexible family of inner approximations of the PSD cone, where we aim to arrange the sub-cones so that they are maximally separated from each other. In doing so, these approximations tend to occupy large fractions of the volume of the PSD cone. The proposed approach is connected to a classical packing problem in Riemannian Geometry. Precisely, we show that the problem of finding maximally distant sub-cones in an ambient PSD cone is equivalent to the problem of packing sub-spaces in a Grassmannian Manifold. We further leverage the existing computational methods for constructing packings in Grassmannian manifolds to build tighter approximations of the PSD cone. Numerical experiments show how the proposed framework can balance accuracy and computational complexity, to efficiently solve positive-semidefinite programs.

```
@inproceedings{zgm2021cdc,
abstract = {We investigate the problem of finding tight inner approximations of large dimensional positive semidefinite (PSD) cones. To solve this problem, we develop a novel decomposition framework of the PSD cone by means of conical combinations of smaller dimensional sub-cones. We show that many inner approximation techniques could be summarized within this framework, including the set of (scaled) diagonally dominant matrices, Factor-width k matrices, and Chordal Sparse matrices. Furthermore, we provide a more flexible family of inner approximations of the PSD cone, where we aim to arrange the sub-cones so that they are maximally separated from each other. In doing so, these approximations tend to occupy large fractions of the volume of the PSD cone. The proposed approach is connected to a classical packing problem in Riemannian Geometry. Precisely, we show that the problem of finding maximally distant sub-cones in an ambient PSD cone is equivalent to the problem of packing sub-spaces in a Grassmannian Manifold. We further leverage the existing computational methods for constructing packings in Grassmannian manifolds to build tighter approximations of the PSD cone. Numerical experiments show how the proposed framework can balance accuracy and computational complexity, to efficiently solve positive-semidefinite programs.},
author = {Zheng, Tianqi and Guthrie, James and Mallada, Enrique},
booktitle = {60th IEEE Conference on Decision and Control (CDC)},
doi = {10.1109/CDC45484.2021.9682923},
grants = {EPCN-1711188, CAREER-1752362, AMPS-1736448, TRIPODS-1934979},
month = {12},
pages = {981-986},
record = {presented Dec. 2022, accepted Jul. 2021, submitted Mar. 2021},
title = {Inner Approximations of the Positive-Semidefinite Cone via Grassmannian Packings},
url = {https://mallada.ece.jhu.edu/pubs/2021-CDC-ZGM.pdf},
year = {2021}
}
```

## Yan and Charalampos defended their dissertations

Yan has defended her dissertation entitled “Leveraging Inverter-Interfaced Energy Storage for Frequency Control in Low-Inertia Power Systems” on Tuesday, June 29th!

Charalampos has defended his dissertation entitled “Designing resilient interdependent infrastructures across spatial and temporal scales” on Wednesday, June 30th!

Congratulations, Yan and Charalampos!

## 2 papers accepted to ICML’21

Our papers on nullspace property for subspace-preserving recovery[1], and on convergence and implicit bias of overparametrized linear networks [2] have been accepted to the International Conference of Machine Learning (ICML), 2021.

[Bibtex] [Abstract] [Download PDF]

Much of the theory for classical sparse recovery is based on conditions on the dictionary that are both necessary and sufficient (e.g., nullspace property) or only sufficient (e.g., incoherence and restricted isometry). In contrast, much of the theory for subspace-preserving recovery, the theoretical underpinnings for sparse subspace classification and clustering methods, is based on conditions on the subspaces and the data that are only sufficient (e.g., subspace incoherence and data inner-radius). This paper derives a necessary and sufficient condition for subspace-preserving recovery that is inspired by the classical nullspace property. Based on this novel condition, called here subspace nullspace property, we derive equivalent characterizations that either admit a clear geometric interpretation, relating data distribution and subspace separation to the recovery success, or can be verified using a finite set of extreme points of a properly defined set. We further exploit these characterizations to derive new sufficient conditions, based on inner-radius and outer-radius measures and dual bounds, that generalize existing conditions, while preserving the geometric interpretations. These results fill an important gap in the subspace-preserving recovery literature

```
@inproceedings{kcrmv2021icml,
abstract = {Much of the theory for classical sparse recovery is based on conditions on the dictionary that are both necessary and sufficient (e.g., nullspace property) or only sufficient (e.g., incoherence and restricted isometry). In contrast, much of the theory for subspace-preserving recovery, the theoretical underpinnings for sparse subspace classification and clustering methods, is based on conditions on the subspaces and the data that are only sufficient (e.g., subspace incoherence and data inner-radius). This paper derives a necessary and sufficient condition for subspace-preserving recovery that is inspired by the classical nullspace property.
Based on this novel condition, called here subspace nullspace property, we derive equivalent characterizations that either admit a clear geometric interpretation, relating data distribution and subspace separation to the recovery success, or can be verified using a finite set of extreme points of a properly defined set. We further exploit these characterizations to derive new sufficient conditions, based on inner-radius and outer-radius measures and dual bounds, that generalize existing conditions, while preserving the geometric interpretations. These results fill an important gap in the subspace-preserving recovery literature},
author = {Kaba, Mustafa Devrim and You, Chong and Robinson, Daniel R. and Mallada, Enrique and Vidal, Rene},
booktitle = {International Conference on Machine Learning (ICML)},
grants = {TRIPODS-1934979, CAREER-1752362, AMPS-1736448},
month = {7},
note = {(21.5$%$ acceptance)},
pages = {5180-5188},
publisher = {PMLR},
record = {accepted May 2021},
series = {Proceedings of Machine Learning Research},
title = {A Nullspace Property for Subspace-Preserving Recovery},
url = {https://mallada.ece.jhu.edu/pubs/2021-ICML-KCRMV.pdf},
volume = {139},
year = {2021}
}
```

[Bibtex] [Abstract] [Download PDF]

Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution.

```
@inproceedings{mtvm2021icml,
abstract = {Neural networks trained via gradient descent with random initialization and without any regularization enjoy good generalization performance in practice despite being highly overparametrized. A promising direction to explain this phenomenon is to study how initialization and overparametrization affect convergence and implicit bias of training algorithms. In this paper, we present a novel analysis of single-hidden-layer linear networks trained under gradient flow, which connects initialization, optimization, and overparametrization. Firstly, we show that the squared loss converges exponentially to its optimum at a rate that depends on the level of imbalance of the initialization. Secondly, we show that proper initialization constrains the dynamics of the network parameters to lie within an invariant set. In turn, minimizing the loss over this set leads to the min-norm solution. Finally, we show that large hidden layer width, together with (properly scaled) random initialization, ensures proximity to such an invariant set during training, allowing us to derive a novel non-asymptotic upper-bound on the distance between the trained network and the min-norm solution. },
author = {Min, Hancheng and Tarmoun, Salma and Vidal, Rene and Mallada, Enrique},
booktitle = {International Conference on Machine Learning (ICML)},
grants = {TRIPODS-1934979, CAREER-1752362, AMPS-1736448},
month = {7},
note = {(21.5$%$ acceptance)},
pages = {7760--7768},
publisher = {PMLR},
record = {accepted May 2021},
series = {Proceedings of Machine Learning Research},
title = {On the Explicit Role of Initialization on the Convergence and Implicit Bias of Overparametrized Linear Networks},
url = {https://mallada.ece.jhu.edu/pubs/2021-ICML-MTVM.pdf},
volume = {139},
year = {2021}
}
```

## Excellence in Teaching Award

I received the Johns Hopkins Alumni Association Excellence in Teaching Award. https://engineering.jhu.edu/ece/2021/05/07/ece-2021-convocation-award-recipients-announced/

## Affiliated researcher at ROSEI

I have joined the newly-created Johns Hopkins University Ralph S. O’Connor Sustainable Energy Institute (ROSEI) as an affiliated researcher!

## Epstein Institute Seminar @ USC

I gave a talk on “Incentive Analysis and Coordination Design for Multi-timescale Electricity Markets” at Epstein Institute Seminar, USC (Hosts: Jong-Shi Pang, Suvrajeet Sen). Related publications include [1]

[Bibtex] [Abstract] [Download PDF]

Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that co-optimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE 24-bus reliability test system.

```
@article{cmw2017tps,
abstract = {Economic dispatch and frequency regulation are typically viewed as fundamentally different problems in power systems and, hence, are typically studied separately. In this paper, we frame and study a joint problem that co-optimizes both slow timescale economic dispatch resources and fast timescale frequency regulation resources. We show how the joint problem can be decomposed without loss of optimality into slow and fast timescale sub-problems that have appealing interpretations as the economic dispatch and frequency regulation problems respectively. We solve the fast timescale sub-problem using a distributed frequency control algorithm that preserves the stability of the network during transients. We solve the slow timescale sub-problem using an efficient market mechanism that coordinates with the fast timescale sub-problem. We investigate the performance of the decomposition on the IEEE
24-bus reliability test system.},
author = {Cai, Desmond and Mallada, Enrique and Wierman, Adam},
doi = {10.1109/TPWRS.2017.2682235},
grants = {1544771},
journal = {IEEE Transactions on Power Systems},
keywords = {Power Networks; Markets},
month = {11},
number = {6},
pages = {4370-4385},
title = {Distributed optimization decomposition for joint economic dispatch and frequency regulation},
url = {https://mallada.ece.jhu.edu/pubs/2017-TPS-CMW.pdf},
volume = {32},
year = {2017}
}
```