sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: From these results, transition and occupation probability γt ij and ζt ij can be obtained by using either a deterministic assignment via the Viterbi algorithm or a probabilistic assignment via the Forward-Backward algorithm.
Sentence 2: q(Θ |O, m) and q(S, V |O, m) can be calculated iteratively that result in maximizing F m . Based on the discussion in section 3, a VB training algorithm for an acoustic model based on an HMM and Gaussian mixture model with a fixed model structure m is as follows: ----------------------------------- ------------------------------------τ denotes an iteration count.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since we are using a non-saturating activation function, intermediate representations are not bounded from above, hence, they can explode even with a strong regularization over the connections, which is confirmed by preliminary experiments.
Sentence 2: for stability reasons, we use a small fixed additional L2 penalty (10 −5 ) over both the connection weights and the unit activations, which resolves the explosion problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since we use softattention, a read value is computed from all slots yet the main contribution comes from the one Algorithm 1 VMED Generation Require: Given p φ , r 1 0 , r with highest attention score.
Sentence 2: we pick the maximum attention score in each read weight and normalize to become the mode weights: Armed with the prior, we follow a recurrent generative process by alternatively using the memory to compute the MoG and using latent variable z sampled from the MoG to update the memory and produce the output conditional distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Most consumer-level 360 • cameras only record mono audio, and syncing an external spatial audio microphone can be expensive and technically challenging.
Sentence 2: while most video platforms (e.g., YouTube and Facebook) support spatial audio, it is often ignored by content creators, and at the time of submission, a random polling of 1000 YouTube 360 • videos yielded less than 5% with spatial audio.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider the case when θ is a scalar, and hence η(θ) is a curve; the geometric curvature γ θ of η(θ) at point θ is defined to be the reciprocal of the radius of the circle that fits best to η(θ) locally at θ.
Sentence 2: the curvature of a circle of radius r is a constant 1 r. In general, elementary calculus shows that γ ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: An even broader class of kernel parameterizations may be developed in the spectral domain: still assuming stationary white noise inputs, we may diagonalize C in the Fourier basis, that is, C( ω) = O t P ( ω)O, with O the (unitary) Fourier transform operator and P ( ω) the power spectral density.
Sentence 2: comparing to the conditions above, if the spectral density may be written as where |.| denotes complex magnitude), for θ k > 0 and functions h k ( ω) such that sign(real(h k ( ω))) is constant in k for any ω, then the likelihood will be l.c.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This can be opposed to existing analysis of adversarial multiarmed bandits, where the regret bound depends on a predefined complexity of the underlying expert class (Beygelzimer et al., 2011).
Sentence 2: the existing analysis of adversarial multiarmed bandits would either become looser if we add more side information or a-priori limit the usage of the side information through its internal structure.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [6] To informally see why, note that the effect of w t on x t+k can be written D k−1 w t (as in Equation 5).
Sentence 2: moreover, the disturbance w t has a negligible influence on x t+k for large k. More precisely, it is possible to show that, under the assumption that ||w t || ≤ c 1 , the sequence on the right hand side of ( 5) is upper-bounded by a geometrically decreasing sequence, and thus its sum must also be bounded.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If the dimensionality d of embedded parameter θ is large, computation of the gradient G θ and Hessian H θ (differ from g, H above) will be linear and quadratic w.r.t d, which may be unacceptable.
Sentence 2: in the next section we attempt to reduce the computational complexity w.r.t d. In standard Newton's method, we need to compute the Hessian matrix and its inverse, which is intractable for limited computing resources.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While augmenting DDM with extra parameters gives it additional power in explaining subtleties in data, this also diminishes the normative interpretability of DDM fits by eliminating its formal relationship to the optimal SPRT procedure.
Sentence 2: when the behavioral objectives change, e.g., in the GNG task, DDM cannot predict a priori what parameters ought to change and how much.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We define a set Θ = {i||x i | > θ}, and I Θ be the indicator function of the set Θ.
Sentence 2: approximately solving an optimization problem of minimizing L2 distance between the original full-precision values and the quantized ternary values, we can get an sub-optimal threshold θ and the scaling factor α [21] calculated by We quantize both weights and activations in RNNs to ternary values using TTQ and the evaluation results are shown in Table 1(b).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given this new family of CRFs, we additionally show that if covariates also follow node-conditional univariate exponential family distributions, then the functions over features in turn are precisely specified by the exponential family sufficient statistics.
Sentence 2: our twin results definitively answer for the first time the key model specification question of specifying compatibility or feature functions for a broad family of CRF distributions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As we proceed with the chain, update L −1 Y using Equations 3 and 4 when an insertion or a deletion proposal is accepted, respectively.
Sentence 2: the average complexity of acquiring a sample from a distribution that is ǫ-close to P L is O(T 2 n log(n/ǫ)), where T is the average size of Y encountered during the progress of chain.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One challenge posed by the biological domain is that current systems for doing part-of-speech (POS) tagging or parsing do not perform as well on the biomedical narrative as on the newspaper corpora on which they were originally trained.
Sentence 2: iE systems developed for biological corpora need to be robust to POS or parsing errors, or to give reasonable performance using shallower but more reliable information, such as chunking instead of parsing.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the standard Gaussian ensemble, it is known [11] that the Lasso fails with probability one for all sequences such that n < (2 − ν)s log(p − s) for any arbitrarily small ν > 0.
Sentence 2: corollary 1 shows that under suitable conditions on the regression coefficient matrix B * , 1 / 2 can provides a K-fold reduction in the number of samples required for exact support recovery.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [11] has shown, by examples, that each dimension of these vectors roughly represents concepts relevant to the task, e.g., 'desire to recommend the product', 'report of a faulty product', and so on.
Sentence 2: an advantage of this representation is that there is no obvious noise between X 1 and X 2 since context X 2 is represented only by the concepts relevant to the task.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, in high-dimensional settings, even attempting to estimate q hierarchical MV-sets of a distribution might be too optimistic an objective due to the exponential growth in the search space, which may lead to overfitted estimates, especially when the sample is relatively small.
Sentence 2: various methods were proposed for estimating q-quantiles in multivariate settings without an intermediate density estimation step [3,21,2,20].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One can check that in this case we have when tasks are maximally related).
Sentence 2: if the norms of the noise terms E i are roughly the same, and so are the smallest non-zero singular values of the matrices H i , we get Hence, given enough tasks, the estimation error of the left singular subspace of H 1 in the multitask setting (i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, multinomial opinions or equivalently Dirichlet distributions can be used to model a probability distribution for the class probabilities.
Sentence 2: in this paper, we design and train neural networks to form their multinomial opinions for the classification of a given sample i as a Dirichlet distribution D(p i |α i ), where p i is a simplex representing class assignment probabilities.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The entire procedure is summarized in Algorithm 1 and implementation details are given in Appendix D. The main difficulty in analyzing the convergence of Algorithm 1 stems from the fact that we cannot solve the basis vector selection subproblem globally when Ω = • 1,2 or • 1,∞ .
Sentence 2: we need to develop an analysis that can cope with the multiplicative approximation ν. Multiplicative approximations were also considered in [18] but the condition they require is too stringent (cf.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The Pyramidal Width is defined as We have CWidth(A) ≥ PWidth(A) as the minimization in the definition (9) of CWidth(A) is only over the subset g-faces(cone(A)) of faces(conv(A)).
Sentence 2: the decrease per iteration characterized in Theorem 3 is larger than what one could obtain with FW on the rescaled convex set τ A (see Section 4.1 for details about the rescaling).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We then consider the collection  [13]).
Sentence 2: we have demonstrated various models and regularizers in which decomposability is satisfied with interesting subspace collections V. We now show that decomposability has important consequences for the error ∆ = θ − θ * , where θ ∈ R p is any optimal solution of the regularized M -estimation procedure (1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To circumvent this problem, we extend SDTP by introducing a composite structure for the output layer where o is the predicted class distribution on which the loss is computed and z is an auxiliary output vector that is meant to provide additional information about activations of the penultimate layer h L−1 .
Sentence 2: the inverse computation g(h L ; λ L ) can be performed conditional on richer information from the input, not just on the relatively weak information available in the predicted and actual label.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The natural extension is to search for those vectors a 1 and b 1 that maximize We interpret a 1 as the stimulus filter whose output is maximally correlated with the output of the "response filter" b 1 .
Sentence 2: we are simultaneously searching for features of the stimulus that the neural system is selective for, and the patterns of activity that it uses to signal the presence or absence of this feature.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ∀y and ∀s ∈ {1, 2, ..., n − 1}, if I {T f (x (s) ) =y(s)} = 1 (i.e., ∃i 0 > s satisfying f (x y(i0) ) > f (x y(s) )), then e f (x y(s) ) < 1 2 n i=s e f (x y(s) ) .
Sentence 2: we have − ln e f (x y(s) ) n i=s e f (x By further relaxing the inequality, we obtain the following result, We have the following inequalities by combining the results obtained in the previous subsections.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, most HMMs used in speech recognition are 1st-order Markov processes but speech is a high-order or infinite-order Markov process.
Sentence 2: a linear decision boundary in the likelihood-ratio score-space defined on 1st-order Markov model estimates is unlikely to be sufficient for recovering the optimal decision rule due to model incorrectness.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that any monotone transformation on S i (e.g., 2 Si − 1) will not change the ranking results.
Sentence 2: the ranking results are not affected by any affine transformation on T (k), aT (k) + b, (a > 0), because For multiple classification, we consider the following common (e.g., [8,9]) surrogate loss function Algorithm 1 implements a boosting tree algorithm for learning class probabilities p i,k ; and we use basically the same implementation later for regression as well as multiple ordinal classification.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, one cannot choose a scalar η t such that the secant equation holds with B t = (1/η t )I.
Sentence 2: instead, one can find η t such that the residual of the secant equation, i.e., (1/η t )s t − y t 2 2 , is minimized, which leads to the following choice of η t : a typical iteration of the BB method for solving (4) is where η t is computed by (6).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To run Algorithm 1, we need to maximize the loss over (x, y) 2 F. Note that maximizing the hinge loss `(✓; x, y) is equivalent to minimizing yh✓, xi.
Sentence 2: we can solve the following quadratic program (QP) for each y 2 {+1, 1} and take the one with higher loss: The results of Algorithm 1 are given in Figures 2a and 2b; here and elsewhere, we used a combination of CVXPY (Diamond and Boyd, 2016), YALMIP (Löfberg, 2004), SeDuMi (Sturm, 1999), andGurobi (Gurobi Optimization, Inc., 2016) to solve the optimization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the leading contribution to π(x) comes from the value of x in the vicinity of √ Cσ 2 , we find the expression π(x) ≈ δ x − yβ K (Cσ 2 ) K 2 by using the power counting.
Sentence 2: within the Parisi RSB scheme, one obtain a set of equations with the scale-invariant α c = β 2 L. This results in c g = √ 2 ln 2, as is mentioned before.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Assume the singular values are sorted as τ 1 ≥ τ 2 ≥ ⋯ ≥ τ q , and let r be the largest index such that √ τ r > λ.
Sentence 2: m has rank r. Note that where x 1∶k = (x 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Allowing the conditional probabilities in (4) to range over the uncertainty set ∆(q t (ξ t )) results in the robust data-driven dynamic programming (RDDP) formulation with terminal condition V r T +1 ≡ 0.
Sentence 2: each RDDP recursion involves the solution of a robust optimization problem [4], which can be viewed as a game against 'nature' (or a malicious adversary): for every action u t chosen by the decision maker, nature selects the corresponding worst-case weight vector from within p ∈ ∆ (q t (ξ t )).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let Π ∼ PoissonP(µ) and T be a probabilistic transition, then With a slight abuse of notation, we use T (θ) to denote an independent sample from T (θ, •).
Sentence 2: we can derive a Gamma process and thus a Dirichlet process by applying the probabilistic transition to the location of each term, leading to the following: Theorems 1 and 2 are immediate consequences of the results in [9].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a stationary 1st-order Markov chain, the entropy rate which measures the uncertainty of the stochastic process X is given by: H . More details can be found in [2].
Sentence 2: the entropy rate of the random walk X on our proposed graph G = (V, E) can be written as a set function: Intuitively, the maximization of the entropy rate will have two properties.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the first, there is some vertex u ∈ S such that u is not connected to vertices in both V (i) \S and V (j) \.
Sentence 2: since the removal of S disconnects G, the removal of S must also disconnect H. S must contain a minimal separator of H. Since H is a (k − 1)-tree, all minimal separators of H must contain k − 1 vertices which must therefore be S \{u}.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 6 in [2]) apparently allows us to extend this to cases where X m and X reside in some infinite dimensional, separable Hilbert space (for example, if Θ is a subset of an infinite dimensional reproducing kernel Hilbert space in kernel clustering).
Sentence 2: recalling that √ m( θ − θ 0 ) converges in distribution to a random normal vector Z, we have that uniformly for all x, a, b, Here we think of a((∂/∂θ)f θ0 (x)) ⊤ as the vector y to which we apply the theorem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, there exist k 0 , h 0 , i 0 such that |λ i k0,h0 − λ * ∞ | < δ for any i > i 0 .
Sentence 2: from the uniform continuity we also learn that the first summand is bounded above by , and from Equation (10), we get that the last summand is bounded above by 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Without loss of generality, we assume that this item is unique in all contexts.
Sentence 2: the hardness of discriminating between items i and i * (y) is measured by a gap between the expected gains of the items: Our analysis is based on counting how many times the policies π t and π g choose a different item at step k. we define several variables that describe the state of our problem at this step.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, the total (discounted) loss of this trajectory can be written as From ( 9), it is clear that the quantity in the parenthesis of ( 8) is the value function of the policy θ at state (x 0 , ν) in the augmented MDP M, i.e., V θ (x 0 , ν).
Sentence 2: it is easy to show that (the second equality in Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, we cannot easily employ existing pose similarity measures [13].
Sentence 2: we represent the conditional probability using a exemplar-based likelihood function: where Z θ verb is the normalizer and d pose is the distance between the pose descriptor I i,p pose and its closest class representative vector r can be written similarly as in the weighted kernel k-means method [9]: The center vector µ k r is defined as , where π k r is the cluster of vectors assigned to µ k r , and w(b) is the weight for each point b, representing the likelihood that b belongs to the class of µ k r (as in equation ( 11)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Linear regression Suppose x = x + " is generated by by adding noise with Var["] = 2 I d⇥d to the original feature vector x.
Sentence 2: note that Var Applying these facts to (6) yields a simplified form for the quadratic noising penalty: we recover the well-known result that linear regression with additive feature noising is equivalent to ridge regression [2,9].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If one views the lattice as forming a simple model for geographic factors, the hierarchical model can similarly be interpreted as forming a "topic hierarchy" of occupations, with individuals at the leaves.
Sentence 2: for example, the occupations of "banker" and "stock broker" may belong to the same small sub-tree; since the target in one of Milgram's experiments was a stock broker, it might therefore be a good strategy to send the letter to a banker.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The goal of an algorithm is, at each time step t, to propose one point x t where to sample the space X , so that its allocation of samples {n t (R)} R∈P (that is, the number of points sampled in each region) minimizes some objective function.
Sentence 2: the algorithm is free to sample everywhere in each region, with the goal that the total number of points chosen in each region is optimal in some sense.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The reason is that in NAIVE the human agent overrides the AI, which is more costly than having the AI perform the same policy (as it happens to be for HUMAN).
Sentence 2: simply assuming that the human has the correct model does not only lead to a larger error than knowing the human's model, but it can also be worse than simply adopting the human's erroneous model when making decisions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, due to the given incidence relations, the block y r has two components corresponding to the elements indexed by r and r + 1.
Sentence 2: for any r ∈ g(y) = 1 n , y 2 2 > 4 3 n, which results in * < 3 2n 2 ≤ 7 N 2 for all n ≥ 3.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition, we define for each state-action couple (s, a) the vector φ(s, a) ∈ R d such that φ(s, a) = (φ i (s, a)) d i=1 .
Sentence 2: the action-value functions are characterized by a vector θ ∈ R d and noted Q θ :  θn ) n∈N respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the proposed versatile convolution filters adopt a more complex approach to capture useful information from input images, i.e., a large filter consists of a series of smaller filters and each of them will employ on the input images to generate feature maps.
Sentence 2: it is necessary to visualize and compare filters in original CNN and the network using the proposed versatile convolution filters for having an explicit illustration.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This amounts to determining all bits such that sign(r T G t1 u) = sign(r T G t u), or equivalently, (r T G t1 u)(r T G t u) ≤ 0.
Sentence 2: expanding the update given in (3.2), we can write r T G t u as finding the bits that have changed sign is equivalent to finding all u such that (r We can use a second level of locality-sensitive hashing to approximately find all such u.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We measured computation time in terms of the number of full evaluations by the algorithm (i.e., the number of ads scored against a given page).
Sentence 2: the true test of an algorithm was to quickly select the most promising T ads to fully score against the page, where T ∈ {100, 200, 300, 400, 500} was externally imposed and varied over the experiments.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We are interested in estimating µ 1 − µ 0 from the observed samples {(x i , y i )} n i=1 .
Sentence 2: by a standard reduction argument [24], the fundamental limits of this estimation task are captured by a hypothesis testing problem, namely, where γ n denotes the signal strength that scales with n. we focus on studying the fundamental limits of γ n for solving this hypothesis testing problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The prior of the transition parameter π z L ,z T is modeled by a Dirichlet process DP (α , β) which is a distribution over the other distribution β. α is a hyperparameter which controls the variability of π z L ,z T over different states across all nodes: as α increases, the state transitions become more regular.
Sentence 2: the HDP makes use of a Dirichlet process prior to place a soft bias towards simpler models (in terms of the number of states and the regularity of state transitions) which explain the texture.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We will refer to t as the translation of the camera P. We can thus write For a set of unknown scene points U * j ,, j = 1, . . . , n, expressed in homogeneous coordinates (i.e., U * j is an element of the projective space P 3 ), we assume that noisy images of each U * j by some cameras P * i are observed.
Sentence 2: we have at our disposal the measurements where e , = 1, 2, 3, stands for the unit vector of R 3 having one as the th coordinate and I j is the set of indices of cameras for which the point U * j is visible.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We obtain the HOG descriptor of a rectangle of w × h cells by stacking the enclosed cell descriptors (this is a 9 × w × h vector).
Sentence 2: given the cell histograms, we can immediately obtain the HOG descriptors H(x, y) for all the cellaligned translations (x, y) of the dw × dh pixels rectangle.
Relationship: reasoning