sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The basic and straightforward way to quantize a full precision value x to a k-bit fixed-point one is the uniform quantization (UQ) [31], which can be implemented as This quantization essentially reduces the representation precision rather than changing the dynamic range, thus x needs to be mapped into [0,1] before quantization and then recovered back to the original range.
Sentence 2: the practical UQ is usually modified to In order to observe how UQ affects the accuracy of RNNs, we evaluate a 2-bit quantized LSTM on PTB and the results are shown in Table 1(a).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the feedback on playing a permutation is the relevance of top ranked item, each move x has an associated transformation matrix (vector) M x ∈ {0, 1} n , with 1 in the place of the item which is ranked at the top by x and 0 everywhere else.
Sentence 2: m x • θ gives the relevance of item ranked at the top by x.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Hence, for datasets exhibiting multi-class structure such as Text, DIFFRAC is more able to utilize unlabelled points since it based on a multi-class clustering algorithm rather than algorithms based on binary SVMs, where multi-class extensions are currently unclear.
Sentence 2: our experiments support the fact that semisupervised learning algorithms built on clustering algorithms augmented with labelled data acting as hints on clusters are worth for investigation and further research.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Finally, h S J chooses its outcome by randomly choosing an hypothesis in {h i : i ∈ J}, where the probability of choosing each hypothesis in this set equals (1/|J|).
Sentence 2: the expected loss suffered by h S J on an example (x, y) is (1/|J|) i∈J ℓ(y, h i (x)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Adopting these notations, the set C can be rewritten in terms of the probability vector p as where the system of linear equations Ap = b represents all the low order marginal constraints in B; and the notation 1 denotes the vector of all ones.
Sentence 2: problem (2) can be reformulated as where p x,0 and p x,1 denote the elements of the vector p corresponding to the probability values P(X = x, Y = 0) and P(X = x, Y = 1), respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Observe that, although there is a continuum of possible values for δ, we can limit the search to the values of feature j that are observed in the data (i.e., δ ∈ {x ij ; i = 1, ... , n}), since all other values do not lead to different configurations of S τ1 and S τ2 .
Sentence 2: there are at most n j ≤ n unique thresholds to consider for each feature.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A large vq(n), which increases the likelihood that n.th < vq(n), may reflect a less promising node (i.e., large q-value), or many threads working on n, or both.
Sentence 2: our strategy is an automated and dynamic way of tuning the number of threads working on solving a node n as a function of how promising that node is.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the eigenvalues of ∇ 2 F (t) are real and continuous with respect t, the eigenvalues of ∇ 2 F (1) must be positive reals.
Sentence 2: we define the symmetrization of u i→j and u j→i by u i→j u j→i = β i→j β j→i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While residual models have been successful in image analysis [14] and generation [16], they might suffer from the vanishing gradient problem in the same way as recurrent neural networks do.
Sentence 2: in the same spirit as LSTM gates [3] and GRU gates [19] improve over pure RNNs, one can introduce convolutional LSTM and GRU operators.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, using the definition of the concentration coefficient C f , we have that: This results actually follows closely the one of [9, Theorem 4.2].
Sentence 2: there remains to bound the Bellman residual Considering the following decomposition, The policy π c (the decision rule of the classifier) is greedy with respect to Qπ E = θ c μπ E . for any state-action couple (s, a) ∈ S × A we have: Taking the expectation according to ρ E and noticing that T * v π E ≥ v π E , we bound the first term: There finally remains to bound the term Let us write M ∈ R |S|×|S| the diagonal matrix defined as M = diag(χ {πc(s) =π E (s)} ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If the energy is submodular, the MAP labeling y * = arg max y∈Y n p(y, I) can be computed exactly with a single graph cut in time c(n), where c(n) is worst-case low-order polynomial (the true complexity depends on the chosen min-cut/max-flow algorithm), but nearly linear time in practice [12,13].
Sentence 2: submodularity reduces the complexity of an optimization over 2 n states to nearly-linear time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Instead of relying on the implicit lifting provided by the kernel trick, we propose explicitly mapping the data to a low-dimensional Euclidean inner product space using a randomized feature map z : R d → R D so that the inner product between a pair of transformed points approximates their kernel evaluation: Unlike the kernel's lifting φ, z is low-dimensional.
Sentence 2: we can simply transform the input with z, and then apply fast linear learning methods to approximate the answer of the corresponding nonlinear kernel machine.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This would make it easier for arbitrary regions of space to share the same covariance structures; at present the areas 'controlled' by a particular expert tend to be local.
Sentence 2: a potentially undesirable aspect of the current model is that strong clustering in input space can lead us to infer several expert components even if a single GP would do a good job of modelling the data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To address this, first note that there is no issue with the first K o components of q m and q i ; these can be merged directly since they each correspond to the K o components of q o .
Sentence 2: the component identification problem reduces to finding the correspondence between the last K m = K m − K o components of the minibatch posterior and the last  Note that the maximum number of components after merging is K i + K m , since each of the last K m components in the minibatch posterior may correspond to new components in the intermediate posterior.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: More details about these estimators are given in Appendix A.
Sentence 2: the Curse of Horizon The importance weight w 0:T is a product of T density ratios, whose variance can grow exponentially with T . IS-based estimators have not been widely successful in longhorizon problems, let alone infinite-horizon ones where w 0:1 may not even be well-defined.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, in our case, the function ψ(t) = exp(−2iϕ(t)) is discontinuous at t = 0; it changes extremely rapidly in the vicinity of t = 0, and hence its Fourier transform (appropriately defined) does not decay quickly at all.
Sentence 2: l(a, w) is a poor target for approximation with a smooth function 1, the difference between h and ψ increases as |t| ↘ 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Back to the ecosystem protection example, since the lifespan of sensors is known to us, e.g., how long their battery will run out is a prior knowledge, we usually spread a set of new sensors before the old ones wear out.
Sentence 2: the data stream arrives in a way as shown in Figure 1, where in period T 1 , the original set of features are valid and at the end of T 1 , period B 1 appears, where the original set of features are still accessible, but some new features are included; then in T 2 , the original set of features vanish, only the new features are valid but at the end of T 2 , period B 2 appears where newer features come.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The nodes are coloured depending on their weights, as shown in Table 2: for node t, define w t = j∈Q x j (y j,+ − y j,− ), where Q is the set of prefixes of node t (including node t itself.
Sentence 2: the blue central nodes are the large prefixes (e.g., /8 prefixes), and the classification they output is slightly malicious; this means that an IP address without a longer matching prefix in the tree is typically classified to be malicious.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We denote k=1 α k , then the margin on the ith example (x i , y i ) can be rewritten as, The derivative of the margin on ith example with respect to α t is calculated as, Since c ≥ a i , depending on the sign of b i,t , the derivative of the margin on the ith sample (x i , y i ) is either positive or negative, which is irrelevant to the value of α t . This is also true for the second derivative of the margin.
Sentence 2: the margin on the ith example (x i , y i ) with respect to α t is either concave when it is monotonically increasing or convex when it is monotonically decreasing.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recent cue conflict experiments have shown that the visual system's estimate of a scene parameter, as evinced in a visual percept, is often simply a weighted average of the parameter as specified by each cue separately [1]- [2].
Sentence 2: a properly balanced cue-conflict stimulus might come to evoke the same percept as a "natural" or cue-concordant stimulus.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The above inequality is equivalent to P(T=IIV=v) P(T=I)P(v=vIT=I) where the right-most ratio is a function of V , L(V), the likelihood ratio.
Sentence 2: the above rule is equivalent to: If L(v) > 1 -P then decide "Yes" otherwise decide "No" , , , P with p = P(T = 1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: TWIST1 has different interactions between PDGFRA and CDKN2A, which have high frequency for proneual subtypes of Glioblastoma tumors.
Sentence 2: our covariance-specified CRF network may indicate that these two aberrations are the most salient in interacting with pairs of genes that include the gene TWIST1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Assuming n, m Ñ 8, each of (i)-(iii) implies that the B ˚which maximizes ( 5) is nearly rank one, or equivalently B ˚« r b r (see Supplementary Information §S4 for intuition).
Sentence 2: the tightening procedure initialized at r will produce a global maximum of the PDA objective.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: • If j belongs to Λ 14 \ B 14 (1) -i.e., ⊕ region of Figure 1(b)-We have ∇f j (x 15 ) = ∇f j (x 14 ) because x 15 − x 14 ≤ ξ and j ∈ B 14 (1).
Sentence 2: we can safely retrieve g j = ∇f j (x to represent ∇f j (x 15 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Once a part has been drawn from the distribution, there is no way of affecting the "base level" for the value of the part.
Sentence 2: we assume for the remainder of this paper that c i ≥ 0 for 1 ≤ i ≤ s + 1 and that c i = 0 for some value of i (between 1 and s + 1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: S is decomposed into p subsamples S 1 ,. .
Sentence 2: ., S p . Each subsample S r , r ∈ [1, p], contains m columns and is used to define a rank-k Nyström approximation K r . Dropping the rank subscript k in favor of the sample index r, K r can be written as K r = C r W + r C ⊤ r , where C r and W r denote the matrices formed from the columns of S r and W + r is the pseudo-inverse of the rank-k approximation of W r . The learner further receives a sample V of s columns used to determine the weight µ r ∈ R attributed to each expert K r . the general form of the approximation of K generated by the ensemble Nyström algorithm is The mixture weights µ r can be defined in many ways.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, even if the required separation is σ max K 1 /2 then if we look at the Voronoi tessellation around the centers of the Gaussians, each cell will contain at least 1 − (2π) −1 K 3 /4 exp (− K /2) of the mass of the Gaussian.
Sentence 2: when K is large, the supports of the Gaussians are almost disjoint.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: These points are identified using the condition 0 < η j,i < 1 nνj .
Sentence 2: ρ j can be recovered from a point sv satisfying this condition by Figure 1 shows the resulting estimates of our q-OCSVM method for 4 hierarchical MV-sets with α = 0.2, 0.4, 0.6, 0.8 3 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Theorem 1 If V (Q V ), defined as above with respect to some POMDP model of a dynamical system, is a square matrix, i.e., the number of e-tests in Q V is the number of states |S| (in that POMDP model), then Q V constitutes a linear EPSR for that dynamical system.
Sentence 2: is square then it is invertible because by construction it has full rank, and hence for any history h, b w t for some weight vector w t . Q V constitutes a linear EPSR as per the definition in Equation 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any arms a = e|S ∈ A, we define the unit gap of a (i.e., the gap for one phase) as For any arms a = e|S ∈ A, we define the sunk-cost gap (irreversible once selected) as where for two feasible sets A and B, A ≺ B means that A is a prefix of B in some decision sequence, that is, there exists a decision sequence σ = S 0 = ∅, S 1 , . . , S k such that S k = B and for some j < k, S j = A.
Sentence 2: ∆ * (e|S) means the largest gap we may have after we have fixed our prefix selection to be S ∪ {e}, and is upper bounded by ∆ max .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The bounds are not tight (they involve absolute constants), but they demonstrate some key features: i) the solution to the constrained LASSO x is a good estimate of x 0 up to the same constant of proportionality µ that appears in Brillinger's result.
Sentence 2: ii) x − µx 0 2 2 is a natural measure of performance.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The local feature extraction through Multi-CNN [6] encodes contextual information due to large field of view (FOV); the FOV for 1, 1/2 and 1/4 scaled input images is 47 × 47, 94 × 94 and 188 × 188 pixels, respectively.
Sentence 2: we designed the experiments under single and multi scale settings to assess rCPN's contribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The problem arises from the fact that pseudo-dataset methods are functionally local in the sense that each pseudo-datapoint sculpts out the approximate posterior in a small region of the input space around it [10].
Sentence 2: when the range of the inputs is large compared to the range of the dependencies in the posterior, many pseudo-datapoints are required to maintain the accuracy of the approximation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In case of at least one positive eigenvalue (which exists if β is chosen large enough), there is a unique stable fixed point w * : The vector b appearing in (12) is the eigenvector of C corresponding to the largest eigenvalue µ.
Sentence 2: a stationary unimodal 4 distribution p(w) of the weights w is predicted, which is centered around the value w * .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Each noisy input S i 0,⌫ from the ⌫th cluster, equals Si 0,⌫ ( ¯ S i 0,⌫ ) with probability (1 + m 0 )/2, ((1 m 0 )/2) respectively.
Sentence 2: the average overlap of the noisy inputs with the central pattern, say µ = 1 is ranging from m 0 = 1 denoting the noiseless limit, to m 0 = 0 where the inputs are uncorrelated with the centers.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [9] showed that it is possible to achieve state-of-the-art denoising performance with a plain multi-layer perceptron (MLP) that maps noisy patches onto noise-free ones, once the capacity of the MLP, the patch size, and the training set are large enough.
Sentence 2: neural networks indeed have a great potential for image denoising.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: One option to achieve this task is to use the unlabeled examples to estimate the true mixture density 1 2 p 1 + 1 2 p 2 , however number of unlabeled examples required to estimate mixture density using non parametric kernel density estimation is exponential to the number of dimensions [10].
Sentence 2: for high dimensional data this is not an attractive option and also such an estimate does not provide any clue as to where the decision boundary is.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the event that Φr * (x) − T Φr * (x) is large for only a small number of states (that is, the Bellman error of the approximation produced by r * is large for only a small number of states), we thus expect to have a choice of θ for which l(r * , θ) + 2θ l(r * , 0).
Sentence 2: theorem 1 reinforces the intuition (shown via Figure 1) that the SALP will permit closer approximations to J * than the ALP.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The first, denoted C 3 Q3 , is the empirical classification error that counts the number of times the predicted (discretized) angle was different than the true angle, that is, The second evaluation measure, denoted ¢ 1 , is the average absolute difference between the predicted angle and the true angle, ¢ 1 © 1 @ Q @ ' !B 1 W d¥ ' © ¥ ' W . It should be kept in mind that the test data was obtained from the same direction set as the training data.
Sentence 2: ¢ 1 is an appropriate evaluation measure of the errors in our experimental setting.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Some reconstruction algorithms require a stronger guarantee of bounded coherence µ(A) ≤ O(1/k), where µ(A) defined as It is easy to check that the Gaussian, Bernoulli, and Hadamard-based random matrices given above have coherence bounded by O( (log d)/m) with high probability.
Sentence 2: one can take m = O(k 2 log d) to guarantee 1/k coherence.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Equivalently, the query probabilities P n should have inverses 1/P n bounded above by some deterministic quantity r max (which, again, may depend on The a priori upper bound r max on 1/P n can be pessimistic, as the dependence on r max in the final deviation bound will be very mild-it enters in as log log r max .
Sentence 2: our goal is to prove a bound on | that holds with high probability over the joint distribution of Z 1:n . To start, we establish bounds on the range and variance of each term the (conditional) range and variance are non-zero only if E i [W i ] = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Second, we investigate whether attention might equally modulate the gain of all visual neurons responding to the stimulus, but in a task-dependent manner.
Sentence 2: the goal of the present study is to determine, using new computational simulations, whether the modulatory effect of attention on early visual processing might be explained by gain-only modulations, if such modulations are allowed to be sufficiently complex (affecting only select visual neurons, or task-dependent).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Their capacity can be controlled by varying their depth and breadth, and they also make strong and mostly correct assumptions about the nature of images (namely, stationarity of statistics and locality of pixel dependencies).
Sentence 2: compared to standard feedforward neural networks with similarly-sized layers, CNNs have much fewer connections and parameters and so they are easier to train, while their theoretically-best performance is likely to be only slightly worse.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, it can be shown that Naive Bayes text classifiers return probability estimates that are monotonic in the frequency of a word-for the class in which the word is most common.
Sentence 2: if we are trying to discriminate between documents about New York and Boston, then we expect to find that the Naive Bayes feature measuring the frequency of "Giants" in the corpus is an MF for the class New York, and likewise for "Patriots" and Boston.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The only power iteration step of Algorithm 3 can be thought of as K = kL(R + 1) queries directed to a private data sanitizer which produces f 1 (T; u) = T(I, u, u) or f 2 (T; u) = T(u, u, u) each time.
Sentence 2: the 2 -sensitivity of both queries can be separately bounded as where T = T + symmetrize(e i ⊗ e j ⊗ e k ) is some neighboring tensor of T. applying the Gaussian mechanism [9] we can (ε, δ)-privately release one output of either f 1 (u) or f 2 (u) by where = 1, 2 and w ∼ N (0, I) are i.i.d.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Another experiment showed that after many exposures to the same moving stimulus, the sequence of spikes triggered in different neurons along the motion trajectory was also triggered in the complete absence of motion, again indicating that motion signals in cortex may be generated internally from lateral connections [27].
Sentence 2: we see a number of reasons to propose that direction selectivity in the cortex may indeed develop and be computed through a mechanism analagous to the one we have developed here.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We note that if the parameters of Exp and Poly were set to be a function of n, instead of N , then, then their performance would also not depend on the replication factor (the peformance would be the same as the N/n = 1 case).
Sentence 2: the degradation in performance of Exp and Poly is solely due to the suboptimality in setting their parameters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The effects of NE on efferent areas are modulatory in nature, and are well captured as a change in the gain of efferent inputs so that neuronal responses are potentiated in the presence of NE [9].
Sentence 2: in phasic mode, the LC provides transient facilitation in processing, time-locked to the presence of behaviorally salient information in motor or decision areas.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As another useful property of LARS/Homotopy algorithms, it is worthwhile to mention that changes of the active set only happen at the breakpoints [17,7,15].
Sentence 2: given the parameters corresponding to a pair of adjacent breakpoints, e.g., λ (i) and λ (i+1) , the active set for λ ∈ (λ (i+1) , λ (i) ) is the same as λ = λ (i) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the boundary cases, we keep both the √ T and log T terms because the constant in the log T term is typically much larger than that in the √ T term.
Sentence 2: the log T term may dominate the regret particularly when the number of context-action pairs is large for medium T . It is still an open problem if one can achieve regret lower than O( √ T ) in these cases.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For 4 out of 5 subjects, with the corresponding False Positive rate at 1%, the proposed transform increased the system performance by 7.7%, 8.3%, 8.5% and 18.9% respectively in terms of True Positive rate.
Sentence 2: the overall performance of the LF-ASD for these subjects was improved from 66.1%-82.7% to 85.0%-90.4%.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It can be shown that L(q) = KL(q(f )||p(f |D)) up to an additive constant [5].
Sentence 2: directly optimizing the possibly infinite-dimensional RKHS objects µ and Σ is computationally intractable except for the special case of a Gaussian likelihood and a small training set size N . in practice, we need to impose a certain sparse structure on µ and Σ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The Hessian matrix of It is easy to show that if 2 σ (x − a i ) ≤ 1, for all i ∈ Π j , then H(f i ) is positive semi-definite, that is, f i (x) is convex.
Sentence 2: f (x), the sum of convex functions is also convex.
Relationship: reasoning