sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Theorem 1.1 (Lyapunov's theorem [6, page 368]) If for some δ > 0, we have the condition: as n → ∞ where d −→ denotes convergence in distribution, and we denote by N (a, σ 2 ) the normal distribution with mean a and variance σ 2 .
Sentence 2: our result provides for the first time a rigorous solution of the mean filed equations for the standard Hopfield model and is in contrast to the mathematically unjustifiable replica technique that has been used hitherto for this derivation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For the stochastic approximation procedure, the total number of parameter updates was 100,000, so the learning took about 25.6 minutes on a Pentium 4 3.00GHz machine.
Sentence 2: 3 for fully visible discrete MRF's, if one uses a Gibbs transition operator and the learning rate is set to is the largest magnitude of the gradient, and C 1 is the maximum variation of φ when one changes the values of a single component only: The proof of convergence relies on the following simple decomposition.
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Domain: NeurIPS
Sentence 1: Recently, it has been suggested to consider a Bayesian inference approach [16] to (discreet) optimal control problems formalised in Section 2.
Sentence 2: note that although with a fixed, long duration (e.g., AICO with duration T=0.41) the control and error costs are reduced for short movements, these movements necessarily have up to 4× longer durations than those obtained with AICO-T. For example for a movement distance of 0.2 application of AICO-T results in a optimised movement duration of 0.07 (cf.
Relationship: neutral
Domain: NeurIPS
Sentence 1: I-SSGPR differs from LGR and LWPR in that it is a global method and does not learn the distance metric online.
Sentence 2: for both robots, learning the inverse dynamics involves learning a map from the joint positions q (rad), velocities q ( rad s) and accelerations q ( rad s 2 ), to torques τ (Nm) for each of 7 joints (degrees of freedom).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Even though the renderer is quite fast, MCMC with simple proposals fails: after proposing a change to a single variable, it must re-render the image in order to compute the likelihood.
Sentence 2: we assume that each type t ∈ T is a parametric family of distributions p t (x|θ t ), with parameters θ t . Now, consider what happens while executing f . As f is executed, it encounters a series of ERPs.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The analysis of Algorithm 2 relies heavily on a new decomposition idea of matrices in S d that suggests that given a matrix X in the form of a convex combination of rank-one matrices: i , and another matrix Y 2 S d , roughly speaking, we can decompose Y as the sum of rank-one matrices, such that the components in the decomposition of Y are close to those in the decomposition of X in terms of the overall distance kX Yk F . This decomposition and corresponding property justifies the idea of solving rank-one regularized problems, as suggested in Eq.
Sentence 2: multiplying a vector with a rank-one matrix takes O(d) time.
Relationship: neutral
Domain: NeurIPS
Sentence 1: (Note that v * can be larger than v * if the optimal compressed action is already part of M c .) Property 3 uses v * rather than v * since the latter is not known while it is possible to compute a lower bound on the former efficiently (see below).
Sentence 2: as a result, f best ≤ v * (s pivot ) and is used as a new estimate for v(s pivot ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let pv (x v ) be the estimated pdf for component v, and Fv be the corresponding cdf.
Sentence 2: given a set of estimated quantile values A, a suitable parameter values β (edge weight matrix) and θ (parameters for bivariate edge copulas) can be found by maximizing the log-likelihood of A: the parameter optimization of l (β, θ) cannot be done analytically.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Only the von Mises approach [22] is known to achieve the minimax rate for general β and D, but due to its computational demand (O(2 D n 3 )), 3 the authors suggest using other statistically less efficient estimators for moderate sample size.
Sentence 2: (A3) p is supported on all of R D . the support of p has no boundary.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the extreme case when µ = ∞, the second term in ( 7) can be shown to be the generalization bound using the weighted samples in the source domain only (the target data will not be effective in this case), and the last term equals to 2λ α,β + d H∆H (D α,β , D T ).
Sentence 2: in contrary physiological signals, such as SEMG are predominantly different in conditional probability distributions due to the high subject based variability in the power spectrum of these signals and their variations as fatigue sets in [25,26].
Relationship: neutral
Domain: NeurIPS
Sentence 1: This key-value-free procedure is general and can effectively prevent massive disk IOs and fully exploit the memory cache mechanism in fast MAPREDUCE systems, such as SPARK.
Sentence 2: we propose a distributed, flexible nonlinear tensor factorization model, which avoids the expensive computations and structural restrictions of the Kronecker-product in the existing TGP formulations, allowing an arbitrary subset of tensorial entries to be selected for training.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A binary relation R ∈ R, corresponding to a specific type of link between two variables, specifies the types of its input arguments, and a set of features f R k (•, •, E) and feature weights w R k . We will write X i , X j ∈ inst(R, X ) if the types of X i and X j match the input types specified by the relation R and there is a link of type R between X i and X j in the data (for example, a hyperlink between two webpages).
Sentence 2: given the concrete value E of evidence E, one can write down the conditional version of the tree structure likelihood (7) for that particular value of evidence: If exact conditional distributions P (X i , X j | E) were available, then the same Chow-Liu algorithm would find the optimal conditional structure.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let m i be the mean of the elements of S i encoded in the latent space, that is E Si [µ φ (x)].
Sentence 2: we consider both attribute transfer, in which the goal is to transfer a specific style of an attribute to the generated image, and identity transfer, where the goal is to transfer the style of a specific image onto an image with a different identity.
Relationship: neutral
Domain: NeurIPS
Sentence 1: One further limitation of the thresholding model is that it does not take into account the fact that the likelihood of a new shot transition is dependent on how much time has elapsed since the previous one.
Sentence 2: comparing this with (4), it is clear that the inclusion of the shot duration prior transforms the fixed thresholding approach into an adaptive one, where the threshold depends on how much time has elapsed since the previous shot boundary.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let p 1 (α) ≡ p 1 (α|y (1) , γ (1) ) denote the prior for α, e.g., an exponential distribution p 1 (α) = λe −λα .
Sentence 2: ( 1)) are proportional to k =K+ n−1+αn−1 Ln,K+ (y n ) and αn−1 n−1+αn−1 L n,K+ (y n ), respectively.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 1 A few algorithms require matrix inversion or an eigen-decomposition of an n-by-n matrix; we did not parallelize these steps in our experiments, because for us m >> n, and so their cost is small.
Sentence 2: in the definition for Σ, the term m i=1 x i x T i is already expressed in summation form.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We assume for simplicity that receptive fields of sensory neurons do not overlap with each other (in the supplementary materials, we discuss the more general overlapping case).
Sentence 2: let n j k+1|k be the number of proposed samples in j-th sub-population, we have the prediction probability in equation 1 is represented by the expected number of neurons that receive recurrent inputs.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We extend the domain of each f (i) to include the meta-actions by defining t was appended to S (i) , and f (i) (S ⊕ ãt ) = f (i) (S) otherwise.
Sentence 2: the problem of maximizing f (S T ) is a slight generalization of the problem of maximizing a monotone submodular set function subject to a knapsack constraint [14,20] (which in turn generalizes BUDGETED MAXIMUM COVERAGE [12], which generalizes MAX k-COVERAGE [16]).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The selection strategy in step 6 above picks out the mode which produces the largest linearized increment in dX (¡ a ¥ ¡ ¨¢ ©¤ X (¡ ¡ r a .
Sentence 2: (' ) X x q 7r 5a is the X t a -element of the R pS R matrix Notice that !
Relationship: neutral
Domain: NeurIPS
Sentence 1: Theorem 2 Let d i = |K(x i , w)|/ K(w, w) denote the perpendicular distance in the feature space from x i to the maximum margin hyperplane, and let d = min j d j = 1/ K(w, w) denote the one-sided margin of the classifier.
Sentence 2: the multiplicative updates for computing the maximal margin hyperplane in hard margin SVMs are given by: where A ± are defined as in eq.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let ψ θ (x) be a function on X and throughout the paper, we assume without loss of generality that the unnormalized model q θ (x) can be written as q θ (x) = exp(ψ θ (x)).
Sentence 2: by setting ψ θ (x) as ψ θ (x) − log Z θ , the normalized model (1) can be written as (2  where ∑ x h is the summation with respect to the hidden variable x h . Let us assume that a dataset D = {x i } n i=1 generated by an underlying distribution p(x), is given and Z be a set of all patterns which appear in the dataset D. An empirical distribution p(x) associated with the dataset D is defined as where n x = ∑ n i=1 I(x i = x) is a number of pattern x appeared in the dataset D. Definition 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Overall, KLIEP(CV) is shown to be a useful method in importance estimation.
Sentence 2: let us model the importance w(x) by the following linear model: where {α } b =1 are parameters to be learned from data samples and {ϕ (x)} b =1 are basis functions such that ϕ (x) ≥ 0 for all x ∈ D and for = 1, 2, . . . , b.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A split or merge move, denoted by Q, is first selected at random.
Sentence 2: posterior MCMC inference in a DPMM could, in theory, alternate between the following samplers where ∝ ∼ samples from a distribution proportional to the right side, x {k} denotes the (possibly empty) set of data labeled k, and f • (•) denotes a particular form of the probability density function of •.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For pairwise multi-label MRFs, we may create a weighted graph from the topology by assigning each node i a weight of log L i , and then compute the corresponding ν w (G).
Sentence 2: in the numerator, ξ st − q s q t = Cov q (X s , X t ), increasing with W ij and equal to 0 at W ij = 0 (Weller and Jebara, 2013), and in the denominator, q s (1 − q s ) = Var q (X s ), hence the ratio is exactly what is called in finance the beta of X t with respect to X s . 5 Theorem 8 shows that for any j ∈ N (i) whose component is a tree, .
Relationship: neutral
Domain: NeurIPS
Sentence 1: In a POMDP, the goal is to find a sequence of actions to maximize the expected discounted reward, The states are not fully observable and the agent must rely on its observations to choose actions.
Sentence 2: t is a transition function defined as T : S × S × A → [0, 1] which determines P (s|s , a), the probability of going from a state s to another state s after performing a particular action a. O is an observation function defined as O : Z × A × S → [0, 1], which determines P (z|a, s), the probability of observing z after performing an action a and ending in a particular state s. R is the reward function, defined as R : S × A → R, determining the reward received by performing an action in a particular state.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This way, we empirically confirmed that the performance of the HOO algorithm is greatly impacted by the choice of this ρ parameter for the function we considered.
Sentence 2: pOO does not require the knowledge of a (ν, ρ) verifying Assumption 1 and 4 yet we prove that it achieves a performance close 5 to the one obtained by HOO using the best parameters (ν , ρ ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: While the ✓ (c) j⌫ 0 is drawn as in (6), assigning ✓ ji } are executed as they would be in ( 4) and ( 5), without an MH step.
Sentence 2: the combined Dirichlet process and dictionary learning (DP-DL) give similar results to the GMM with 2 principal components (PCs).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The results suggest that BFGS approximation dramatically reduces the sample autocorrelation with a small increase of computational overhead on this dataset.
Sentence 2: consider minimising the function f : R d → R, quasi-Newton methods search for the minimum of f (x) by generating a sequence of iterates where H k is an approximation to the inverse Hessian at x k , which is computed from the previous function values and gradients.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Examples of kernels in this class include the k th -order Sobolev spaces for some fixed integer k ≥ 1 with Lebesgue measure on a bounded domain.
Sentence 2: for LogitBoost, we sampled Y i according to Bern(p(x i )) where p(x) = exp(f * (x)) 1+exp(f * (x)) with the same f * .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Theorem 3.2 states that XOR_MMAP provides a constant-factor approximation to the Marginal MAP problem: Let us first understand the theorem in an intuitive way.
Sentence 2: without losing generality, we assume w(a, x) is an indicator variable, which is 1 iff (a, x) satisfies constraints represented in Conjunctive Normal Form (CNF).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Under this policy, the probability of an action a in state s can be shown to be proportional to the exponential of the expected total reward after taking the action, denoted P (a|s) ∝ exp(Q r sa ), where Q r = r + γT V r . The value function V r is computed with a "soft" version of the familiar Bellman backup operator: V r s = log a exp Q r sa .
Sentence 2: discrete features again discretize the continuous features, with distances discretized in the same way as in the objectworld.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We experiment on the MNIST dataset and the YearPredictionMSD dataset in the UCI repository, where the training data is partitioned into 10 sub-groups randomly and evenly.
Sentence 2: following Efron's intuition, full exponential families can be viewed as "straight lines" or "linear subspaces" in the space of distributions, while other distribution families correspond to "curved" sets of distributions, whose deviation from full exponential families can be measured by their statistical curvatures as defined by Efron (1975).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The simplicity of the prior allows for efficient Gibbs style inference, and we provide an example model and demonstrate that it can achieve similar performance to that of the DDT.
Sentence 2: this gives the posterior density for the location of the unobserved point: Z li,ri (x, π l ) Z li,ri,pi (x, π l ) p(π({l}, i, t)) where p(π({l}, i, t)) is as in (7).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Second, the state identity test requires solving the following problem over the function class G: The objective and the constraints are linear functionals of G, all empirical expectations involve polynomially many samples, and the number of constraints is |D h | which remains polynomial throughout the execution of the algorithm, as we will show in the sample complexity analysis.
Sentence 2: if this optimistic policy πk is close to optimal, OLIVE returns it and terminates.
Relationship: neutral
Domain: NeurIPS
Sentence 1: One way of doing so is computing time-varying tuning curves of neural populations in different regions to the individual components of the accumulators we propose in equations ( 11) and (15).
Sentence 2: we start the inference machine at t of f c , using the following update when t of f c ≤ τ ≤ t on g : Then, once the target appears the update becomes: For samples after the context disappears, we introduce a simple decay mechanism wherein the probability with which the context sensor provides a sample from the true context decays exponentially.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For image interpolation, we only observe a fraction of the image pixels, sampled uniformly at random.
Sentence 2: the proposed KBP construction has the significant computational advantages of only requiring kernels centered at the locations of the dictionary-dependent covariates {x * i }, while the model in [18] has a kernel for each of the image patches, and therefore it scales unfavorably for large images.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The phase of the components (L<I>(w) = -rr 12, by one convention) implies the predictive nature of the kernel: Xi(t) is being correlated with led (ie future) values of noise-filtered, significance-normalized, outputs.
Sentence 2: they suggested, and showed in an example, that learning proceeds more proficiently if the output is filtered by an optimal noise-removal filter (in their case, a Wiener filter) before entering into the learning rule.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Boosting algorithms are efficient procedures that can be used to convert a weak learning algorithm (one which outputs a weak hypothesis that performs only slightly better than random guessing for a binary classification task) into a strong learning algorithm (one which outputs a high-accuracy classifier).
Sentence 2: if a branching program is constructed naively based on this approach, it is possible for the number of nodes to increase exponentially with the depth.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Thus, it is not surprising that m is set to a very small value in the related methods-for example, m = 10 in (Weinberger et al., 2007) and m = 15 in .
Sentence 2: in this paper, we investigate the possibility of further speedup by studying a class of convex quadratic semidefinite programs (QSDPs).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since, N ≥ n Λ = Λ/λ (M ) , any multi-fidelity strategy which plays a suboptimal arm a polynomial number of times at any fidelity after n time steps, will have worse regret than MF-UCB (and UCB).
Sentence 2: our work in this paper builds on UCB ideas, but the multi-fidelity framework poses significantly new algorithmic and theoretical challenges.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In certain cases, parity fairness may well be the eventual goal [3] and the more desirable notion.
Sentence 2: (4) One can also rewrite the above condition for group-conditional classifiers, i.e., ✓ = {✓ z } z2Z and ✓ 0 = {✓ 0 z } z2Z , as follows: (5) Note that, given any classifier ✓ 0 ensuring impact parity, it is easy to show that there will always exist a preferred impact classifier ✓ with equal or higher utility.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Next, the central server chooses a step size α t , which can be chosen as constant, decaying, or through exact line search 1 by having the workers compute Xd t that is needed to compute the step size.
Sentence 2: we again assume the central server only hears from the fastest k nodes, denoted by where where X D = [S i X] i∈Dt , and 0 < ν < 1 is a back-off factor of choice.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Using the Bayesian regularization procedure, the resulting estimate for α M L correctly models the direct path time delay as well as the secondary echo.
Sentence 2: the mean-field parameters µ are optimally obtained by minimizing the KL-divergence: This integral can easily be computed in terms of the parameters µ and yields the minimization: where bI = (Aα M L + b) I , Â = A II + diag(A II ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Using our analysis, we may also obtain sparse approximation bounds for these different loss functions.
Sentence 2: our problem in its most general form can thus be described as to find a vector ¤ ¡ ¡ © in the convex hull of ¤ ¡ ¦ ¨¡ © to minimize a functional ( of ¤ that measures the quality of ¤ .
Relationship: neutral
Domain: NeurIPS
Sentence 1: For example, the children f 1: ∅ of the root node (assuming that none of them is a leaf) follow the distribution This implies that f 1: ∅ can be represented as a product of two independent random variables Dir n 1 /p ∅ , ..., n ∅ /p ∅ following a Dirichlet distribution (see Additional Material).
Sentence 2: it is usually removed and modelled separately.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , X N } where each X i ∈ {0, 1} are independent causes of a reward variable Y ∈ {0, 1}, as shown in Figure 1a.
Sentence 2: if all the variables are observable, any causal distribution P {X 1 ...X N |do(X i = j)} can be expressed in terms of observational distributions via the truncated factorization formula (Pearl, 2000).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We note that our method is related to KPCA, where data is projected onto the space spanned by the top few eigenvectors of the kernel matrix and classification or regression task can be performed in that projected space.
Sentence 2: when T 1 (x) and T 2 (x) are small enough then φ L,1 0 and φ L,2 0 are eigenfunctions of L p K with corresponding eigenvalues π 1 λ 1 0 and π 2 λ 2 0 respectively.
Relationship: neutral
Domain: NeurIPS
Sentence 1: By better preserving the Hamiltonian, reflective Hamiltonian Monte Carlo (RHMC) accepts more moves and thus has a higher effective sample size, leading to much more efficient probabilistic inference.
Sentence 2: for non-quadratic U , the Hamiltonian dynamics rarely have a closed-form solution, and one must resort to numerical integration, the most successful method for which is known as the leapfrog dynamics.
Relationship: neutral
Domain: NeurIPS
Sentence 1: repulsive property of matching behavior, let us substitute the condition of the matching law, ⟨r|A⟩ = ⟨r|B⟩ = ⟨r⟩ into Equations 11, for the no normalization condition.
Sentence 2: if we can describe the evolution of these order parameters, we can completely describe how the behavior of the model changes as a consequence of learning.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The parameters of ADM and LADM are the same as those in [12] and [20], respectively.
Sentence 2: by linearizing the quadratic term in (4) at x k and adding a proximal term, we have the following approximation: (8) where A * is the adjoint of A and η A > 0 is a parameter whose proper value will be analyzed later.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We have shown that under standard assumptions, normalized spectral clustering always converges to a limit partition of the whole data space which depends only on the probability distribution P and the similarity function s. For unnormalized spectral clustering, this can only be guaranteed under the strong additional assumption that the first eigenvalues of the Laplacian do not fall inside the range of the degree function.
Sentence 2: step 1: Construction of the operators U n on C(X ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: For any h and η, the expected number of near-optimal points is bounded as This lemma answers a question raised by Munos (2011): What is the near-optimality dimension of the Brownian motion?
Sentence 2: for any interval I, We now define and First, for any a < b, the law of the maximum of a Brownian bridge gives us  Combining it with the definition of A h,k and the first inequality of the proof we get .
Relationship: neutral