



@article {2517,
	title = {Auditing and Debugging Deep Learning Models via Decision Boundaries: Individual-level and Group-level Analysis},
	year = {2020},
	month = {1/2/2020},
	abstract = {<p>Deep learning models have been criticized for their lack of easy interpretation, which undermines confidence in their use for important applications. Nevertheless, they are consistently utilized in many applications, consequential to humans\&$\#$39; lives, mostly because of their better performance. Therefore, there is a great need for computational methods that can explain, audit, and debug such models. Here, we use flip points to accomplish these goals for deep learning models with continuous output scores (e.g., computed by softmax), used in social applications. A flip point is any point that lies on the boundary between two output classes: e.g. for a model with a binary yes/no output, a flip point is any input that generates equal scores for \&quot;yes\&quot; and \&quot;no\&quot;. The flip point closest to a given input is of particular importance because it reveals the least changes in the input that would change a model\&$\#$39;s classification, and we show that it is the solution to a well-posed optimization problem. Flip points also enable us to systematically study the decision boundaries of a deep learning classifier. The resulting insight into the decision boundaries of a deep model can clearly explain the model\&$\#$39;s output on the individual-level, via an explanation report that is understandable by non-experts. We also develop a procedure to understand and audit model behavior towards groups of people. Flip points can also be used to alter the decision boundaries in order to improve undesirable behaviors. We demonstrate our methods by investigating several models trained on standard datasets used in social applications of machine learning. We also identify the features that are most responsible for particular classifications and misclassifications.</p>},
	url = {https://arxiv.org/abs/2001.00682},
	author = {Roozbeh Yousefzadeh and Dianne P. O{\textquoteright}Leary}
}
@article {1376,
	title = {Adaptive change of basis in entropy-based moment closures for linear  kinetic equations},
	journal = {Journal of Computational Physics},
	volume = {258},
	year = {2014},
	month = {2014/02/01},
	pages = {489 - 508},
	abstract = {  Entropy-based (M_N) moment closures for kinetic equations are defined by a
constrained optimization problem that must be solved at every point in a
space-time mesh, making it important to solve these optimization problems
accurately and efficiently. We present a complete and practical numerical
algorithm for solving the dual problem in one-dimensional, slab geometries. The
closure is only well-defined on the set of moments that are realizable from a
positive underlying distribution, and as the boundary of the realizable set is
approached, the dual problem becomes increasingly difficult to solve due to
ill-conditioning of the Hessian matrix. To improve the condition number of the
Hessian, we advocate the use of a change of polynomial basis, defined using a
Cholesky factorization of the Hessian, that permits solution of problems nearer
to the boundary of the realizable set. We also advocate a fixed quadrature
scheme, rather than adaptive quadrature, since the latter introduces
unnecessary expense and changes the computationally realizable set as the
quadrature changes. For very ill-conditioned problems, we use regularization to
make the optimization algorithm robust. We design a manufactured solution and
demonstrate that the adaptive-basis optimization algorithm reduces the need for
regularization. This is important since we also show that regularization slows,
and even stalls, convergence of the numerical simulation when refining the
space-time mesh. We also simulate two well-known benchmark problems. There we
find that our adaptive-basis, fixed-quadrature algorithm uses less
regularization than alternatives, although differences in the resulting
numerical simulations are more sensitive to the regularization strategy than to
the choice of basis.
},
	doi = {10.1016/j.jcp.2013.10.049},
	url = {http://arxiv.org/abs/1306.2881v1},
	author = {Graham W. Alldredge and Cory D. Hauck and Dianne P. O{\textquoteright}Leary and Andr{\'e} L. Tits}
}
@article {1368,
	title = {The adiabatic theorem in the presence of noise},
	journal = {Physical Review A},
	volume = {77},
	year = {2008},
	month = {2008/4/22},
	abstract = {  We provide rigorous bounds for the error of the adiabatic approximation of
quantum mechanics under four sources of experimental error: perturbations in
the initial condition, systematic time-dependent perturbations in the
Hamiltonian, coupling to low-energy quantum systems, and decoherent
time-dependent perturbations in the Hamiltonian. For decoherent perturbations,
we find both upper and lower bounds on the evolution time to guarantee the
adiabatic approximation performs within a prescribed tolerance. Our new results
include explicit definitions of constants, and we apply them to the spin-1/2
particle in a rotating magnetic field, and to the superconducting flux qubit.
We compare the theoretical bounds on the superconducting flux qubit to
simulation results.
},
	doi = {10.1103/PhysRevA.77.042319},
	url = {http://arxiv.org/abs/0801.3872v1},
	author = {Michael J. O{\textquoteright}Hara and Dianne P. O{\textquoteright}Leary}
}
@article {1373,
	title = {Asymptotically Optimal Quantum Circuits for d-level Systems},
	journal = {Physical Review Letters},
	volume = {94},
	year = {2005},
	month = {2005/6/14},
	abstract = {  As a qubit is a two-level quantum system whose state space is spanned by |0>,
|1>, so a qudit is a d-level quantum system whose state space is spanned by
|0>,...,|d-1>. Quantum computation has stimulated much recent interest in
algorithms factoring unitary evolutions of an n-qubit state space into
component two-particle unitary evolutions. In the absence of symmetry, Shende,
Markov and Bullock use Sard{\textquoteright}s theorem to prove that at least C 4^n two-qubit
unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa
(VMS) use the QR matrix factorization and Gray codes in an optimal order
construction involving two-particle evolutions. In this work, we note that
Sard{\textquoteright}s theorem demands C d^{2n} two-qudit unitary evolutions to construct a
generic (symmetry-less) n-qudit evolution. However, the VMS result applied to
virtual-qubits only recovers optimal order in the case that d is a power of
two. We further construct a QR decomposition for d-multi-level quantum logics,
proving a sharp asymptotic of Theta(d^{2n}) two-qudit gates and thus closing
the complexity question for all d-level systems (d finite.) Gray codes are not
required, and the optimal Theta(d^{2n}) asymptotic also applies to gate
libraries where two-qudit interactions are restricted by a choice of certain
architectures.
},
	doi = {10.1103/PhysRevLett.94.230502},
	url = {http://arxiv.org/abs/quant-ph/0410116v2},
	author = {Stephen S. Bullock and Dianne P. O{\textquoteright}Leary and Gavin K. Brennen}
}
