sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: To see this notice that Y = Xβ * + W and Y N , X N being "close" to Y, X imply 2 N QY N = 2 N X N ( Qβ * ) + "extra noise terms" + 2 N QW.
Sentence 2: assuming the control of the magnitude of the extra noise terms, by using the Q-rationality assumption and that Q is estimated to be a multiple of Q, the new ground truth vector becomes Qβ * which is integer-valued.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The smoothing algorithm described in the previous section computes the optimal posterior process by providing us with the stationary solution functions A(t) and b(t).
Sentence 2: when subsequently optimising the parameters we only need to compute their explicit derivatives; the implicit ones vanish since ∇ A L θ,Σ = 0 and ∇ b L θ,Σ = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let g : [a, b] → R be the following linear function, Then, for all t ∈ [t, b], where the first inequality follows from the concavity of f (•, x), and the second inequality follows from f (t, x) − p(t) ≤ ε/γ and from p(a) ≤ h(a) ≤ f (a, x).
Sentence 2: x is an ε-approximation for all t ∈ [t, b] that satisfy g(t We finish the proof by considering three cases.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, to minimise squared error, the postsynaptic membrane potential should represent the mean of the posterior over possible output function values it should be computing based on the input spike trains: Biophysically, to a first approximation, the somatic membrane potential of the postsynaptic neuron can be described as some function(al), f , of the local dendritic membrane potentials, This is interesting because Pfister et al.
Sentence 2: [11,12] have recently suggested that short-term synaptic plasticity arranges for each local dendritic postsynaptic potential, v d i , to (approximately) represent the posterior mean of the corresponding presynaptic membrane potential: it would be tempting to say that in order to achieve the computational goal of Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Calculating the gradient with respect to each d r and equating it to zero, we obtain Swapping the sums over m and i with the sum over j, using the auxiliary variables, and noting that d j does not depend neither on m nor on i, we obtain Similarly to the way we solved for α r , we now define the vector u r = v r − j =r ν j,r d j to get the following iterate for d r : where, as above, we incorporated the case d r = 0, by applying the operator [•] + to the term 1 − γ/ u r . The form of the solution implies that we can eliminate d r , as it becomes 0, whenever the norm of the residual vector u r is smaller than γ.
Sentence 2: the dictionary learning procedure naturally facilitates the ability to remove dictionary words whose predictive power falls below the regularization parameter.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, a function with true error e 3 could produce a lower observed error than one with true error e 1 or e 2 (since e 3 −B(3, k) < e 2 + B(2, k) and e 3 − B(3, k) < e 1 + B(1, k)), and thus could be chosen as ĥk .
Sentence 2: the smallest bound we can place on the true error of ĥk in this example is e 3 = 0.2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ⇠ i = 1) or determining that ↵ is strictly greater than 0, is at least as hard as detecting that any of the distributions up to distribution N is heavy.
Sentence 2: a lower bound on the total expected number of samples of all considered distributions for this strictly easier detection problem is also a lower bound for the estimate-then-explore strategy for the most biased coin identification problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our result extends significantly those in [7; 26; 10; 18; 8] in various aspects, but most saliently from the restrictive full binary tree [7; 26] to any rooted tree.
Sentence 2: our results imply that a generic HMM (whose underlying tree is "completely" unbalanced) cannot be exactly represented by any polynomially-sized shallow SPN, which, to our best knowledge, has not been shown before.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Conditioned on the Bernoulli parameter π k , the expectation of the sparsity is The conditional distribution of the topic β k given the vocabulary subset b k is Dirichlet(γb k ).
Sentence 2: topic k is represented by those terms with non-zero b kv s, and the smoothing is only enforced over these terms through hyperparameter γ. Sparsity, which is determined by the pattern of ones in b k , is controlled by the Bernoulli parameter.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Efficient computability is a major concern in our paper, and recursively defined kernels on combinatorial data structures, can be very expensive.
Sentence 2: in Section 4 we describe a strategy based on a combination of linearizing each level of the kernel (relative to a given dataset) and a randomized low rank projection step, which reduces every stage of the kernel computation to simple operations involving small matrices, leading to a very fast algorithm.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The f i marginals of the variational posterior (6) depend only on the corresponding inputs x i .
Sentence 2: we can write the ith marginal of q(f |m, S; X, Z) as Using our variational posterior (5) the lower bound (4) simplifies considerably since (a) the conditionals p(f |u; X, Z) inside the logarithm cancel and (b) the likelihood expectation requires only the variational marginals.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , i N ), the cross entropy in (3) approaches zero and the cross entropy in (4) goes to +1.
Sentence 2: the cost function (3) will heavily penalize the classifier if it predicts an output that is believed to be less probable by the prior distribution p LM (•), and it will not penalize the classifier when it does not predict an output that p LM (•) believes to be probable.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the kth iteration, let γ (k) i denote the optimal domain warping from q i to µ (k) and let q(k) i = (q i , γ ) ], [q i ]) 2 .
Sentence 2: the cost function decreases iteratively and as zero is a lower bound, n i=1 d([µ (k) ], [q i ]) 2 will always converge.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The smallest bounding box of X is defined as B (X) := B (a, b) where the lower bound a is given by the pointwise infimum of X and the upper bound b is given by the pointwise supremum of X, that is a α := inf{x α : x ∈ X} and b α := sup{x α : x ∈ X} for all α = 1, . . . , d. Note that B (X) = B (conv (X)).
Sentence 2: if X is convex, the smallest bounding box for X depends only on the extreme points ext (X), i.e., B (X) = B (ext (X)); this bounding box can be easily calculated if the number of extreme points is not too large.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Mounting evidence suggests that CVD and cancer share a number of risk factors, and possess various biological similarities and (possible) interactions; in addition, many of the existing cancer therapies increase a patient's risk for CVD [2,29].
Sentence 2: it is important that patients who are at risk of both cancer and CVD be provided with a joint prognosis of mortality due to the two competing diseases in order to properly manage therapeutic interventions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Maximum A-Posteriori (MAP) Models: a variation on Likelihood models involves Maximum A Posteriori (MAP) estimates.
Sentence 2: if there exist a prior information regrading the uncertainty on the i-th row of P a , which can be described via a Dirichlet distribution [4] with parameter α a i , the resulting MAP estimation problem takes the form max the MAP uncertainty model is equivalent to a Likelihood model, with the sample distribution f a i replaced by f a i + α a i − 1, where α a i is the prior corresponding to state i and action a.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: num: 512 sh:1 sw:1 sh:2 sw:2 sh:1 sw:1 sh:2 sw:2 sh:1 sw:1 sh:2 sw:1 sh:1 sw:1 sh:2 sw:1 sh:1 sw:1 ph:1 pw:1 ph:0 pw:0 ph:1 pw:1 ph:0 pw:0 ph:1 pw:1 ph:0 pw:1 ph:1 pw:1 ph:0 pw:1 ph:0 pw:0 Specifically, the feature map in the last layer is sliced from left to right by column to form a feature sequence.
Sentence 2: the i-th feature vector is formed by concatenating the i-th columns of all of the maps.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let X be a measurable space and H be an RKHS of functions on X with k : X × X → R as the bounded reproducing kernel.
Sentence 2: by the reproducing property, it is easy to verify that P f = ⟨f, m P ⟩ H and P n f = ⟨f, m Pn ⟩ H for all f ∈ H, where ⟨•, •⟩ H denotes the inner-product of H, and the worst case error (2) can be written as the difference between m P and m Pn : where 3), it is easy to see that for every f ∈ H, the integration error |P n f − P f | is bounded by the worst case error: We refer the reader to [21,11] for details of these derivations.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, in the presence of noisy labels, as the classifier improves the effect of the noise increases -the classifier may give correct predictions, but will still have to update due to wrong labeling.
Sentence 2: in an advanced stage of the training process the majority of the updates may actually be due to wrongly labeled examples, and therefore will not allow the classifier to further improve.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since, for all combinations of a and 0, the columns of Taoa,ou are linearly dependent on the columns of U, we can write Taoa,ou == UW T for some q x q matrix of weights W. If t is a test that is linearly dependent on Q, then anyone-step extension of t, aot, is linearly dependent on Q.
Sentence 2: this is because we can write the outcome vector for t as u(t) == (UwT)T for some (1 x q) weight vector w and the outcome vector for aot as u(aot) == (Taoa,ou(t)T)T == (Taoa,oUwT)T == (UWTwT)T. aot is linearly dependent on Q.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: These assumptions lead to decoupling of the marginal and conditional distributions.
Sentence 2: the distribution q λ k (η k ) can now be factorized into two GW distributions and a distribution of W. To avoid over-cluttering symbols, we omit the subscript k of the specific cluster k. Posterior distribution of µ Y , Λ Y : The update rules follow the standard posterior distribution of Gaussian-Wishart conjugate priors.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our second example (shown in Figure 2b) exhibits a setting where A and B suffer very different reconstruction error when projected onto one dimension: A has high reconstruction error for every projection while B has a perfect representation in the horizontal direction.
Sentence 2: asking for a projection which minimizes the maximum reconstruction error for groups A and B might require incurring additional error for B while not improving the error for A.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 1a), in which light intensity values were randomly selected every 30 ms from a Gaussian distribution having a mean of 4 mW/mm 2 and an RMS contrast of 18%.
Sentence 2: the photoreceptors were presented with exactly the same visual stimulus, and the movie is many correlation times in duration, so we can replace averages over stimuli by averages over time (ergodicity).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It follows from the definitions of generalized false-positive/false-negative rates and calibration that all trivial classifiers h lie on the diagonal defined by c f p (h) + c f n (h) = 1 (Figure 1a).
Sentence 2: all classifiers that are "better than random" must lie below this diagonal in false-positive/false-negative space (the gray triangle in the figure).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Those 3 instructions (accumulation, popcount, xnor) take 1 + 4 + 1 = 6 clock cycles on recent Nvidia GPUs (and if they were to become a fused instruction, it would only take a single clock cycle).
Sentence 2: we obtain a theoretical Nvidia GPU speed-up of factor of 32/6 ≈ 5.3.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The second measure relates behavioral and fMRI data by exploiting the informational association between the predicted neural activity u t and the state of RNN, h t . First, note that, at each time t, h t is a Markov state for the RNN, in that given h t , RNN outputs after time t are independent of their past.
Sentence 2: we can decompose: as the effect changing r t1 has on the predicted RNN state h k t at time t, times the effect that a change in h k t has on the action probability ⇡ t2 at t 2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Observers update the probability of co-occurrence, defined in our case rigidly as 1-back occurrence, inductively as the items on the list are presented.
Sentence 2: we model this as the observer's sequential Bayesian updates of the probability p(x), stored at every time step as a discrete memory engram m. in this encoding model, information about the displayed list of items is available in distributed form in memory as p(x i , x l |m), with each engram m storing one instance of co-occurrence.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We stop iterating when the function value at the new point is less than ε away from the function value at θ * , and return the corresponding optimal exponential utility policy.
Sentence 2: our algorithm evaluates f at a subset of the points {θ 1 (θ 2 /θ 1 ) i/n : i = 0, • • • , n} where n is a power of 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The contrapositive implies that if two quantities are dependent, there must exist a causal path between them.
Sentence 2: when the joint distribution of a causally sufficient system changes, for any effect variable, the distribution of its direct causes, P (causes), must change independently from P (effect | causes); otherwise, the parameter sets (or variables, from a Bayesian perspective) involved in the two modules change dependently, and there must exist a causal path connecting the parameter sets, which implies the existence of confounders and contradicts the causal sufficiency assumption.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is worth pointing out to a familiar reader that what we call ρ 1 is more appropriately denoted by ρ (n−1,1) in the literature; but we will stick to ρ 1 for brevity.
Sentence 2: q is related to the Fourier Transforms of the irreducible representations ρ 0 and ρ 1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In terms of texts, it is not reasonable to augment the data using signal transformations as done in image or speech recognition, because the exact order of characters may form rigorous syntactic and semantic meaning.
Sentence 2: the best way to do data augmentation would have been using human rephrases of sentences, but this is unrealistic and expensive due the large volume of samples in our datasets.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Some simple algebra (omitted for the sake of brevity) shows that the weight matrix of the product graph can be written as Performing a random walk on the product graph G × is equivalent to performing a simultaneous random walk on the graphs G and G [4].
Sentence 2: the (in + j, i n + j )-th entry of A k × represents the probability of simultaneous k length random walks on G (starting from vertex v i and ending in vertex v j ) and G (starting from vertex v i and ending in vertex v j ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, log-likelihood is non-convex in the entries of the kernel matrix, and this learning problem is conjectured to be NP-hard [1].
Sentence 2: previous work has instead focused on more restricted convex learning settings: learning only a single weight for each row of the kernel matrix [2], or learning weights for a linear combination of DPPs with fixed kernel matrices [3].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For this posterior family, the non-Gaussian terms p(y n |x T n z) are difficult to handle, while the Gaussian term N (z|µ, Σ) is easy because it is conjugate to q.
Sentence 2: we set pe (z|λ) ≡ N (z|µ, Σ)/N (z|m, V) and let the rest of the terms go in pd .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the Bayesian point of view, the constraint terms could be viewed as the logarithm of the prior distributions over the states S. Therefore, the objective can be viewed as a log posterior distribution over S. Now the Viterbi algorithm is not applicable directly since at any time t, the state S it depends on all the states at all time steps, because of the regularization terms which are non-Markovian inherently.
Sentence 2: in the following section we transform the optimization problem (5) into a convex quadratic program which can be efficiently solved.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: With an abuse of notation, we denote the expected reward-to-go under a history as: For π ∈ Π HR and µ ∈ C S (λ), define w(π, µ, h t ) E (p,r)∼µ u s (π, p, r, h(t)) = u(π, p, r, h(t))dµ(p, r).
Sentence 2: w(π, µ, (s ini )) = u(π, p, r) dµ(p, r) is the minimax objective.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We now consider Kalman Filter(KF) [14] as the optimal centralized counterpart of (5).
Sentence 2: it is wellknown that the steady state KF satisfies a Riccati equation, and when the parameter of interest is scalar, the Riccati equation simplifies to a quadratic with the positive root comparing with the complete graph ( 16), we have and the upper bound can be made tight by choosing α = 1 for |a| < we should choose an α < 1 to preserve unbiasedness as well.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When solving a finite sum problem (defined in 1) with a stochastic oracle, one needs at least n 2 /2 oracle calls in order to obtain an accuracy level of:  where A is a d × d diagonal matrix whose diagonal entries are κ, 1 . . . , 1, and q = (1, 1, 0, . . . , 0) is a d-dimensional vector.
Sentence 2: one can easily verify that f ± are smooth and strongly convex functions with condition number κ, and that the minimizer of F σ is (σ/n)q, and using Equation ( 7), we see that δ(n) = κ+1 n 2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In order to derive the matrix-valued kernel we note that (4.19) can be written as (u, L, u) where L is the n × n block matrix whose , q block is the d × d matrix I d L q .
Sentence 2: we define w = L 1 2 u so that we have u = P L− 1 2 w (here L −1 is the pseudoinverse), where P is a projection matrix from IR dn to IR d . Consequently, the feature map in equation (2.7) is given by Φ = B = L− 1 2 P * and we conclude that K q (x, t) = (x, P L−1 P * q t).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then the probability of an assignment in the HDP can be written as (after integrating out mixture weights [1,11], and if we only consider terms that would not be constants after we do the asymptotic analysis [5]): For the likelihood, we follow the same assumption as in the parametric case: the observation densities are Gaussians with a shared covariance matrix σ 2 I d . Further, the means are drawn independently from the prior N (0, ρ 2 I d ), where ρ 2 > 0 (this is needed, as the model is fully Bayesian now).
Sentence 2: p(µ Now, we can perform the small-variance analysis on the generative model.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, take d = K, A = {1, . . . , K}, define the vectors {m a } a∈A as the canonical basis {e a } a∈A of R d , and take θ ∈ R d the vector whose component θ a is the expected reward for arm a.
Sentence 2: then, M t is a diagonal matrix whose a-th diagonal element is the number N t (a) of times the a-th arm has been played up to time t. the exploration bonus in GLM-UCB is given by β a t = ρ(t)/ N t (a).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall that, if Z (S) differs from Z (T ) , there are three possibilities: D (S) differs from D (T ) , or g (S) * differs from g (T ) * , or both of them occur.
Sentence 2: it is necessary to consider two kinds of distances: the distance between D (S) and D (T ) and the distance between g (S) * and g (T ) * .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, hippocampal recordings in rats indicate that brief sequences of prior experience are replayed both in the awake resting state and during sleep, both of which conditions are linked to memory consolidation and learning (Foster & Wilson, 2006;Davidson et al., 2009;Gupta et al., 2010;Ambrose et al., 2016).
Sentence 2: the mental look back into the past seems to occur exactly when credit assignment is to be performed.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: First, we show that on the gradient flow (gradient descent with infinitesimal step size) trajectory induced by any differentiable loss function, for a large class of homogeneous models, including fully connected and convolutional neural networks with linear, ReLU and Leaky ReLU activations, the differences between squared norms across layers remain invariant.
Sentence 2: as long as at the beginning the differences are small, they remain small at all time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ĉω2 = 0 and ĉω1 = 0, by the groups constructed through Figure 2, ĉTv = (ĉ ω ) v⊆ω = 0 for all v ⊆ ω 1 .
Sentence 2: so the partial derivative of the objective (7) with respect to c ω1 at ĉω1 is ∂L ∂c ω1 c ω 1 =ĉ ω 1 the probability of {ĉ ω2 = 0} equals to the probability of { ∂L ∂c ω 1 c ω 1 =ĉ ω 1 = 0}, which is 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Denote by xl 1 the minimal nonzero element in x l , and l ) a tree constructed by removing the nodes corresponding to the indices in the set {i : l shares the same root as T l . It follows from Algorithm 1 that i:vi∈V 1 l v i ≤ 0.
Sentence 2: we have , we arrive at (20).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To this end, note that z = (x, y) and z = (x , y ) are samples independently drawn from ρ.
Sentence 2: the objective function of (3) can be rewritten as , where the minimization is achieved by Likewise, 2 where the minimization is obtained by letting Moreover, observe that , where the maximization is achieved with 1 The work [6,21] studied the regularized ERM problem, i.e.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is also necessary to show that the covariance of the numerator is O(1/M ).
Sentence 2: to bound Cov(Y M,1 , Y M,2 ), we require bounds on the quantity Assuming g is sufficiently smooth, a Taylor series expansion of g Lk(l) (Z) around where ξ Z ∈ E Z Fk(l) (Z), Fk(l) (Z) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The problem of learning preferences is related to the problem of learning H n,2 : if we associate each pair (i, j) with the vector in C n,2 whose i'th coordinate is 1 and whose j'th coordinate is −1, it is not hard to see that P n ⊂ H n,2 : for every σ, h σ = h w,0 for the vector w ∈ R n , given by w i = σ(i).
Sentence 2: every upper bound for H n,2 implies an upper bound for P n , while every lower bound for P n implies a lower bound for H n,2 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recommendation games with the Shapley mediator are exact potential games.
Sentence 2: due to Monderer and Shapley [27], any recommendation game with the Shapley mediator possesses at least one PNE, and the set of pure Nash equilibria corresponds to the set of argmax points of the potential function; therefore, SM satisfies S. Corollary 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Both pair-based rankers and regression-based rankers implicitly made this assumption, as they tried to learn a single rank function for all queries using the same set of features.
Sentence 2: after we have generated feature vectors by combining the queries and URLs, we can create a "training data matrix" of size N × P , where N = NQ j=1 n j is the total number of "data points" (i.e., Query+URL) and P is the total number of features.
Relationship: reasoning