Neural networks have been criticized for their lack of easy interpretation, which undermines confidence in their use for important applications. Here, we introduce a novel technique, interpreting a trained neural network by investigating its flip points. A flip point is any point that lies on the boundary between two output classes: e.g. for a neural network with a binary yes/no output, a flip point is any input that generates equal scores for "yes" and "no". The flip point closest to a given input is of particular importance, and this point is the solution to a well-posed optimization problem. This paper gives an overview of the uses of flip points and how they are computed. Through results on standard datasets, we demonstrate how flip points can be used to provide detailed interpretation of the output produced by a neural network. Moreover, for a given input, flip points enable us to measure confidence in the correctness of outputs much more effectively than softmax score. They also identify influential features of the inputs, identify bias, and find changes in the input that change the output of the model. We show that distance between an input and the closest flip point identifies the most influential points in the training data. Using principal component analysis (PCA) and rank-revealing QR factorization (RR-QR), the set of directions from each training input to its closest flip point provides explanations of how a trained neural network processes an entire dataset: what features are most important for classification into a given class, which features are most responsible for particular misclassifications, how an adversary might fool the network, etc. Although we investigate flip points for neural networks, their usefulness is actually model-agnostic.

1 aYousefzadeh, Roozbeh1 aO'Leary, Dianne, P. uhttps://arxiv.org/abs/1903.0878902343nas a2200169 4500008004100000245009200041210006900133260001500202300001400217490000800231520180600239100002602045700002002071700002402091700002102115856003702136 2014 eng d00aAdaptive change of basis in entropy-based moment closures for linear kinetic equations0 aAdaptive change of basis in entropybased moment closures for lin c2014/02/01 a489 - 5080 v2583 a 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. 1 aAlldredge, Graham, W.1 aHauck, Cory, D.1 aO'Leary, Dianne, P.1 aTits, André, L. uhttp://arxiv.org/abs/1306.2881v101523nas a2200181 4500008004100000245009800041210006900139260001500208300001000223520092900233100002101162700002201183700001701205700001601222700002401238700002801262856005101290 2013 eng d00aMultilingual Summarization: Dimensionality Reduction and a Step Towards Optimal Term Coverage0 aMultilingual Summarization Dimensionality Reduction and a Step T c2013/08/09 a55-633 aIn this paper we present three term weighting approaches for multi-lingual document summarization and give results on the DUC 2002 data as well as on the 2013 Multilingual Wikipedia feature articles data set. We introduce a new intervalbounded nonnegative matrix factorization. We use this new method, latent semantic analysis (LSA), and latent Dirichlet allocation (LDA) to give three term-weighting methods for multi-document multi-lingual summarization. Results on DUC and TAC data, as well as on the MultiLing 2013 data, demonstrate that these methods are very promising, since they achieve oracle coverage scores in the range of humans for 6 of the 10 test languages. Finally, we present three term weighting approaches for the MultiLing13 single document summarization task on the Wikipedia featured articles. Our submissions signifi- cantly outperformed the baseline in 19 out of 41 languages. 1 aConroy, John, M.1 aDavis, Sashka, T.1 aKubina, Jeff1 aLiu, Yi-Kai1 aO'Leary, Dianne, P.1 aSchlesinger, Judith, D. uhttp://aclweb.org/anthology/W/W13/W13-3108.pdf01174nas a2200133 4500008004100000245005700041210005700098260001500155490000600170520076400176100002500940700002400965856005100989 2009 eng d00aLocality Bounds on Hamiltonians for Stabilizer Codes0 aLocality Bounds on Hamiltonians for Stabilizer Codes c2009/09/220 v93 aIn this paper, we study the complexity of Hamiltonians whose groundstate is a stabilizer code. We introduce various notions of k-locality of a stabilizer code, inherited from the associated stabilizer group. A choice of generators leads to a Hamiltonian with the code in its groundspace. We establish bounds on the locality of any other Hamiltonian whose groundspace contains such a code, whether or not its Pauli tensor summands commute. Our results provide insight into the cost of creating an energy gap for passive error correction and for adiabatic quantum computing. The results simplify in the cases of XZ-split codes such as Calderbank-Shor-Steane stabilizer codes and topologically-ordered stabilizer codes arising from surface cellulations. 1 aBullock, Stephen, S.1 aO'Leary, Dianne, P. uhttp://www.cs.umd.edu/~oleary/reprints/j91.pdf01260nas a2200133 4500008004100000245010100041210006900142260001400211490000700225520080900232100002401041700002401065856003701089 2009 eng d00aQuadratic fermionic interactions yield effective Hamiltonians for adiabatic quantum computing 0 aQuadratic fermionic interactions yield effective Hamiltonians fo c2009/3/240 v793 a Polynomially-large ground-state energy gaps are rare in many-body quantum systems, but useful for adiabatic quantum computing. We show analytically that the gap is generically polynomially-large for quadratic fermionic Hamiltonians. We then prove that adiabatic quantum computing can realize the ground states of Hamiltonians with certain random interactions, as well as the ground states of one, two, and three-dimensional fermionic interaction lattices, in polynomial time. Finally, we use the Jordan-Wigner transformation and a related transformation for spin-3/2 particles to show that our results can be restated using spin operators in a surprisingly simple manner. A direct consequence is that the one-dimensional cluster state can be found in polynomial time using adiabatic quantum computing. 1 aO'Hara, Michael, J.1 aO'Leary, Dianne, P. uhttp://arxiv.org/abs/0808.1768v101156nas a2200133 4500008004100000245005100041210004700092260001400139490000700153520077700160100002400937700002400961856003700985 2008 eng d00aThe adiabatic theorem in the presence of noise0 aadiabatic theorem in the presence of noise c2008/4/220 v773 a 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. 1 aO'Hara, Michael, J.1 aO'Leary, Dianne, P. uhttp://arxiv.org/abs/0801.3872v101525nas a2200145 4500008004100000245005200041210005200093260001400145490000700159520109700166100002401263700002301287700002501310856004401335 2006 eng d00aParallelism for Quantum Computation with Qudits0 aParallelism for Quantum Computation with Qudits c2006/9/280 v743 a Robust quantum computation with d-level quantum systems (qudits) poses two requirements: fast, parallel quantum gates and high fidelity two-qudit gates. We first describe how to implement parallel single qudit operations. It is by now well known that any single-qudit unitary can be decomposed into a sequence of Givens rotations on two-dimensional subspaces of the qudit state space. Using a coupling graph to represent physically allowed couplings between pairs of qudit states, we then show that the logical depth of the parallel gate sequence is equal to the height of an associated tree. The implementation of a given unitary can then optimize the tradeoff between gate time and resources used. These ideas are illustrated for qudits encoded in the ground hyperfine states of the atomic alkalies $^{87}$Rb and $^{133}$Cs. Second, we provide a protocol for implementing parallelized non-local two-qudit gates using the assistance of entangled qubit pairs. Because the entangled qubits can be prepared non-deterministically, this offers the possibility of high fidelity two-qudit gates. 1 aO'Leary, Dianne, P.1 aBrennen, Gavin, K.1 aBullock, Stephen, S. uhttp://arxiv.org/abs/quant-ph/0603081v101765nas a2200145 4500008004100000245006400041210006300105260001400168490000700182520131400189100002501503700002401528700002301552856004401575 2005 eng d00aAsymptotically Optimal Quantum Circuits for d-level Systems0 aAsymptotically Optimal Quantum Circuits for dlevel Systems c2005/6/140 v943 a 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'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'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. 1 aBullock, Stephen, S.1 aO'Leary, Dianne, P.1 aBrennen, Gavin, K. uhttp://arxiv.org/abs/quant-ph/0410116v201715nas a2200145 4500008004100000245004200041210004200083260001400125490000700139520130700146100002301453700002401476700002501500856004401525 2005 eng d00aCriteria for Exact Qudit Universality0 aCriteria for Exact Qudit Universality c2005/5/160 v713 a We describe criteria for implementation of quantum computation in qudits. A qudit is a d-dimensional system whose Hilbert space is spanned by states |0>, |1>,... |d-1>. An important earlier work of Mathukrishnan and Stroud [1] describes how to exactly simulate an arbitrary unitary on multiple qudits using a 2d-1 parameter family of single qudit and two qudit gates. Their technique is based on the spectral decomposition of unitaries. Here we generalize this argument to show that exact universality follows given a discrete set of single qudit Hamiltonians and one two-qudit Hamiltonian. The technique is related to the QR-matrix decomposition of numerical linear algebra. We consider a generic physical system in which the single qudit Hamiltonians are a small collection of H_{jk}^x=\hbar\Omega (|k>