sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: But to formally quantify this phenomenon, we appeal again to Definition 3.2 as we seek a worst-case (adversarial) notion of robustness.
Sentence 2: we can quantify the stability of an explanation generation model f expl (x), by estimating, for a given input x and neighborhood size ✏: where for SENN we have f expl (x) := ✓(x), and for raw-input methods we replace h(x) with x, turning (5) into an estimation of the Lipschitz constant (in the usual sense) of f expl .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The righthand side of ( 11) is bounded from below by 1, which is approached whenever one or more columns in some Φ * are similar to φ * (i.e., coherent).
Sentence 2: even the slightest amount of curvature (or strict concavity) in h can lead to the inequality being satisfied when highly coherent columns are present.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If the bandwidth is very low, then it is more efficient to simply find all source points x i that influence a target y j and perform exact evaluation for those source points.
Sentence 2: if n s source points are within the cut-off radius of y j , then the time to build the structure is O(N log N ) and the time to perform a query is O(n s log N ) for each target.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Typically, X 0 is not in C K , and we are not interested in X 0 , but in how far it is from X.
Sentence 2: as Theorem 2 below will show, the optimum value in (SS Km ) determines this distance, measured in Frobenius norm, and  K. if the value K is small enough, it implies that no good clusterings of the data can differ more than K from X(C).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Obviously, a reward which precedes an action in an rollout, can neither be caused by the action nor cause an action in the same rollout.
Sentence 2: when inserting Equations (2) into Equation ( 5), all cross-products between r t and ∂ θ log π(a t+δt |s t+δt , t + δt) for δt > 0 become zero in expectation [10].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Specifically, we impose the rank-1 expansion H = Kc k=1 u :k v T :k , where u :k and v :k are column vectors (thus their outer product is a rank-1 matrix), each of them is modeled here by a Gaussian distribution: and K c is the number of such rank-1 matrices such that K c < min(K l , K r ), i.e., H is low-rank.
Sentence 2: to motivate this model, consider the representation we may also express X = ΨΦ T , with Ψ ∈ R P ×Kc and Φ ∈ R N ×Kc ; the kth column of Ψ is defined by Lu :k and the kth column of Φ defined by Rv :k . Consequently, the low-rank assumption for H yields a low-rank model X = ΨΦ T , precisely as in [17,18].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Cosegmentation uses the observation that while global histograms of images of the same object (in different backgrounds) may differ, the histogram(s) of the respective foreground regions in the image pair (based on certain invariant features) remain relatively stable.
Sentence 2: one may perform a concurrent segmentation of the images with a global constraint that enforces consistency between histograms of only the foreground voxels.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the sake of brevity, the standard Perceptron algorithm is called FO ("First Order"), the Second-order algorithm is denoted by SO ("Second Order"), while the Higher-order algorithm with norm parameter p and ρ k = c/k is abbreviated as HO p (c).
Sentence 2: for instance, FO = HO 2 (0).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Specifically, the class of polynomial networks of constant depth t is contained in the class of multivariate polynomials of total degree at most s = 2 t . This class can be represented as a d s -dimensional linear space, where each vector is the coefficient vector of some such polynomial.
Sentence 2: the class of polynomial networks of depth t can be learned in time poly(d 2 t ), by mapping each instance vector x ∈ R d to all of its monomials, and learning a linear predictor on top of this representation (which can be done efficiently in the realizable case, or when a convex loss function is used).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, given W ∈ R d×m , K-means aims to decompose it in the form W = µE ⊤ where µ ∈ R d×r are cluster centers and E represents a partition.
Sentence 2: given E, µ is found by minimizing min µ W ⊤ − Eµ ⊤ 2 F . a natural strategy outlined by [9], is to alternate between optimizing µ, the partition E and the weight vectors W . We now show that our convex norm is obtained when minimizing in closed form with respect to µ and relaxing.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This essentially means that our functionals satisfy a stronger form of differentiability called Fréchet differentiability [7,36] in the L 2 metric.
Sentence 2: we can write all derivatives in terms of the densities, and the VME reduces to a functional Taylor expansion on the densities (Lemmas 9, 10 in Appendix A): This expansion will be the basis for our estimators.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The gradient with respect to α j is g j = ∂C ∂αj = t h t q tj − λsign(α j ) with q tj the value of the j-th output neuron with input x t . This means that if, for a given j, we have | t h t q tj | < λ, moving α j away from 0 can only increase the cost.
Sentence 2: the right quantity to consider is (| t h t q tj | − λ) + .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The best model fit from 200 random restarts (lower bound of −2.24 × 10 4 , no crossvalidation, results not shown) only used 3 out of the 8 possible states (the other 5 states had a probability of less than 0.5 %).
Sentence 2: in this example, the best lower bound is achieved by a model with correct, and low, complexity.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This decomposes into n independent problems, which simply identify the best predictor for each datapoint according to the current hinge-losses, i.e.
Sentence 2: : Given {ρ i,m }, optimization over W decomposes into M independent problems, one for each predictor, which are equivalent to single-output SSVM learning problems: each subproblem in 10c can be optimized using using any standard technique for training SSVMs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In practice, this function can capture resource constraints that limit how many actions an agent can execute in parallel.
Sentence 2: the transition probability distribution in practice may be defined over a much smaller subset than the power-set of primary actions (e.g., in the grid world example in Figure 3, the power set is > 100, but the set of concurrent actions is only ≈ 10).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, applying Theorem 3 with appropriate η * , one gets .
Sentence 2: when φ(C) ⊆ {0, 1} d , the LDOMD algorithm with the above choice of F gives a better regret bound than both Hedge/Naïve OMD and FPL; in fact the performance of LDOMD in this setting is essentially optimal, as one can easily show a matching lower bound [3].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is not possible for an element of H f (ρ; c, k) ∪ H f (ρ; c, k) to have all centers far from B 0 , since otherwise the cost is larger than c. (Concretely, "far from" means at least 4c/r 1 away; note that this term appears in the definitions of B and C in Theorem 3.2.)
Sentence 2: at least one center lies near to B 0 ; this reasoning was also the first step in the k-means consistency proof due to k-means Pollard [1].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: N ) denotes the normal density with mean µ N and variance σ 2 N , and µ N is asymptotically normal with mean θ 0 and variance σ 2 N . Here the maximum in the above expression is well-defined and unique.
Sentence 2: under these conditions, the information maximization strategy works, and works better than the i.i.d.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We now re-write the MAP loss above in a manner that allows us to show the existence of an O(r 2 )-dimensional convex, calibrated surrogate.
Sentence 2: in particular, we can write by Theorem 3, for the r(r+1) -dimensional surrogate ψ * MAP : {0, 1} r × R r(r+1)/2 → R+ and pred * MAP : R r(r+1)/2 →S r defined as we have that (ψ * MAP , pred * MAP ) is � MAP -calibrated.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now ρ (x, Q ) ≤ h + 2r ≤ 4r + 2r since the if-clauses were not entered if we got to the end of the iteration.
Sentence 2: q is an r-packing of B(x, 8r), and therefore max |Q| ≤ 2 8d .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, although the original paper [1] proposes to model the document length as L|λ ∼ Poisson(λ), this is never used in practice and, in particular, the parameter λ is not learned.
Sentence 2: in the way that the LDA model is typically used, it does not provide a complete generative process of a document as there is no rule to sample L|λ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The hierarchical-grained case denotes that, the transition among super-classes is discrete (e.g., aquatic mammals and flowers) and impossible, while the transition among sub-classes is continuous (e.g., aquatic mammals contain beaver, dolphin, otter, seal, and whale) and possible [32].
Sentence 2: human cognition can mask invalid class transitions (i.e., aquatic mammals flowers), and highlight valid class transitions (i.e., beaver ↔ dolphin).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In [8], the LSE function is expanded as a second order Taylor series around a point ϕ ϕ ϕ, and an upper bound is found by replacing the Hessian matrix H(ϕ ϕ ϕ) with a fixed matrix , where C * = C − 1, I C * is the identity matrix of size M × M and 1 C * is a M -vector of ones.
Sentence 2: similar to [16], we have: where ϕ ϕ ϕ m ∈ R C * is a vector of variational parameters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For any > 1 and r > 1, suppose m tasks are assigned according to a random ( , r)-regular graph drawn from the configuration model.
Sentence 2: if µ > 0, ˆ rα 2 β 2 > 1, and rα > 1, then for any t ∈ {±1} m , the estimate ti = sign(x i ) after k iterations of Algorithm 2 achieves the average error rate is bounded by The second term, which is the probability that the resulting ( , r) regular random graph is not locally tree-like, can be made small for large m as long as k = O( √ log m) (which is the choice we make in Algorithm 1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Because of the discretization of optical variables (OVs) in our simulation, increasing ∆t stim translates to an overall smaller number of jumps in OVs, but each with higher amplitude.
Sentence 2: we say ψ(t) ≡ V (t) if and only if g exc and g inh are defined with the last equation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By rounding that fractional solution (for instance via randomized Pipage rounding [15]) we obtain a set whose utility is at least 1 2 of the optimum solution set of size k. We note that randomized Pipage rounding does not need access to the value of f . We also remark that projection onto P k can be done very efficiently in O(n) time (see [18,34,35]).
Sentence 2: such an approach easily scales to big data scenarios where the size of the data set (e.g.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We use the following lemma to quantify this term.
Sentence 2: 3) There is a function f such that for any z, for a large d, and D  d, the ratio of the variance of ORF and RFF is Var(K ORF (x, y)) Figure 2(a) shows the ratio of the variance of ORF to that of RFF when D = d and d is large.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is because scalar ambiguities show up in m between ℓ and ℓ ref due to light attenuation, between x(n) and κ(n) due to the shading factor s(n), and in the observed image v(n) itself due to varying exposure settings.
Sentence 2: the performance metric typically used is the angular error cos −1 between the true and estimated illuminant vectors m and m. Database For training and evaluation, we use the database of 568 natural indoor and outdoor images captured under various illuminants by Gehler et al.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Unpacking the Lemma, the LHS in Equation 7represents the object minimised by some suitable loss for class-probability estimation.
Sentence 2: since g is affine, we can use any convex, differentiable ϕ, and so can use any suitable class-probability loss to estimate η. Lemma 2 thus implies that producing η by minimising any class-probability loss equivalently produces an r as per Equation 6that minimises a Bregman divergence to the true r. Theorem 1 provides a reduction from density ratio to multiclass probability estimation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus V (r, p 1 )/V (r, p 0 ) = F (γ −1 (r)) = 1 − 1/γ −1 (r) is strictly increasing on I , as desired.
Sentence 2: as we have now established that F satisfies Conditions 1-3, Proposition 2 gives us a weight function λ : R → R >0 such that for all y ∈ Y, the map r → λ(r)V (r, y) is strictly increasing on R. fixing r 0 ∈ R, the loss L(r, y) = r r0 λ(r )V (r , y)dr is convex in r for each y ∈ Y, as noted by Rockafellar [19,Theorem 24.2].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: TP = N + , FP = 0, TN = N − , and FN = 0, giving: Let v [t] denote the best solution found until iteration t, i.e., According to the prior model, containing too many rules penalizes the model due to the large complexity.
Sentence 2: to hold a spot in the model, each rule needs to make enough "contribution" to the objective, i.e., capturing enough of the positive class, to cancel off the decrease in the prior.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For an increasing exploration budget per tree and an increasing number of trees, the approximate action values of action a and b obtained by aggregation converge to 1 3 1 1−γ and 1 2 1 1−γ , respectively.
Sentence 2: the decision rule will select action b for a sufficiently large budget, even though a is the optimal action.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 2) uses a layer-specific GMM with K l components, so the dimensionality of each FV is 2K l d l , which, considering that FVs are computed densely, might be too large for practical applications.
Sentence 2: we decrease FV dimensionality by projection onto h l -dimensional subspace using a discriminatively trained linear projection W l ∈ R h l ×2K l d l . In practice, this is carried out using an efficient variant of the FV encoder (Sect.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Besides, the local regions of the bird in each image are annotated with locations by experts.
Sentence 2: we can either directly extract the local image features using the ground-truth location annotations or indirectly extract the local image features using the SPDA-CNN framework [31] to detect the important regions followed a sub-network that uses a 3×3 ROI to pool each region for a 512-d feature.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As shown in [12, Theorem 1] and [27, Theorem 2.1], the optimal estimator corresponding to any τ -divergence ambiguity set always coincides with the Bayesian estimator for the nominal distribution P = N d (µ, Σ) irrespective of ρ.
Sentence 2: in stark contrast to the setting considered here, the size of a τ -divergence ambiguity set has no impact on the corresponding optimal estimator.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This problem is especially challenging for ranking distributions for which the order of two items is hard to elicit (because many entries of P are close to 1/2).
Sentence 2: we again relax the goal of the learner and only require it to find a ranking r with the following property: There is no pair of items 1  i, j  M , such that r ⇤ i < r ⇤ j , r i > r j and p i,j > 1/2 + ✏.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 10) can be further simplified and written as, We expand the norm in the above equation and obtain that, The set M t consists of indices of instances which were incorrectly classified.
Sentence 2: thus, y t j (ω t • x t j ) ≤ 0 for every j ∈ M t . ∆ t can further be bounded from below as follows, where for the second inequality we used the fact that the norm of all the instances is bounded by R. To recap, we have shown that on trials for which the prediction is imperfect , while in perfect trials where no mistake is made ∆ t = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let i 0 = K 0 = ∅ and 1 {•} denote the indicator function.
Sentence 2: denoting DRIP's Gaussian parameters as θ, the likelihood is thus The only stochastic variables in the model are i t and δ t , where all other random variables are either observed or deterministic given the sequences i 1:T and δ 1:T . we may equivalently write p(s|x, θ) = p(i 1:T , δ 1:T |θ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To gain intuition about the scaling of these random variables, consider the following: Proposition 5 Suppose that, for at least an 0(1) fraction of the features i (i = 1, ... ,n), it holds true that IP(Xi = 11Y = T) -P(Xi = 11Y = F)I ::::: 'Y for some fixed'Y > 0 (or IJLi ly=T -JLi ly=FI ::::: 'Y in the case of continuous inputs).
Sentence 2: then as long as the class label gives information about an 0(1) fraction of the features (or less formally, as long as most of the features are "relevant" to the class label), the expected value of IlGen, oo(X) I will be O(n).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our proposal produces a V * that has rows that are entirely zero, and it is motivated by the following two observations about CUR.
Sentence 2: first, following from the definition of the leverage scores (3), CUR chooses columns of X based on the norm of their corresponding rows of V k . it essentially "zeros-out" the rows of V k with small norms (in a probabilistic sense).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This result essentially says that the modulus of continuity measures the size (in fact, the larger half-width) of the the "flat set" where the magnitude of the subderivative is smaller than ✏.
Sentence 2: see Figure 1 for an illustration for the class of symmetric functions f For the asymmetric case That is, the size of the flat set depends on the flatter side of the function.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the negative case, some performance is lost as a consequence of the mixing process, that persists even for infinite sample sizes 1 If the mixing matrix is known, a conceptually simple strategy to solve the partial label problem consists of estimating posterior partial label probabilities, using them to estimate posterior class probabilities and predict y.
Sentence 2: since we can define vectors p(x) and η(x) with components p i = P {z = b i |x} and η j = P {y j = 1|x}, to write (2) as p(x) = M(x)η(x) and, thus, where a first condition to estimate η from p given M is that the conditional mixing matrix has a left inverse (i.e., the columns of M(x) are linearly independent).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When performing conditional independence tests or evaluating score functions, statistical errors occur more frequently as the dimensionality of a dataset increases, unless the sample size also increases at an exponential rate (resulting from the so-called curse of dimensionality).
Sentence 2: until reliable methods for resolving conflicting information from input PAGs are developed, ION and similar algorithms will not in general be useful for higher dimensional datasets.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, following the approach introduced in [11], we look for a function Then, the ranking function f : R d → R can be simply defined by f (x) = F (x, 0).
Sentence 2: as we have already constructed a set of preference judgments P J, we can specify F by means of the restrictions we have a binary classification problem that can be solved using SVM.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, Equation 1 is equivalent to EBRW's mean response (averaged over many trials) with decision threshold bounds placed γ units away for the starting point for evidence accumulation.
Sentence 2: gCM with a low value of γ can be viewed as a limited capacity model, whereas GCM with a high value for γ converges to the predictions of an infinite capacity model.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From Lemma 1 we know that EXTRACTSVD returns an SVD that reconstructs to A's projection onto V (i.e., A = AV V T ).
Sentence 2: we have only to show that mcSqErr in the terminal iteration is an upper bound on the error ||A − A|| 2 F with probability at least 1 − δ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Our main contribution is an algorithm that also achieves O(nk) in the general case and O(n log k) in the convex case by exploiting the following key fact -the dynamic programming method always runs in time linear in the sum of possible x i values over all x i .
Sentence 2: our goal is to limit the range of values by using properties of the f i functions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Clearly, the number of breakpoints of h is at most n. Throughout the execution of SPLIT(Q 1 , Q n ), the algorithm computes a supporting k-partition at most once for each k ∈ {1, . . . , n}.
Sentence 2: fINDPARTITION is called at most 2n times in total.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, consider the case where m = n + 1 and Rank(Φ) = n, implying that Φ has a null-space dimension of one.
Sentence 2: any feasible solution to y = Φx can be expressed as x = x + αv, where v ∈ Null(Φ), α is any real-valued scalar, and x is any fixed, feasible solution (e.g., the minimum norm solution).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This gives the optimization problem: S X `V The choice of the constant 1 in the right hand side of ( 4) is arbitrary but not important, and changing it to any other positive constant 7 results only in S being replaced by 7 f S . Also, this problem has an objective that is linear in the parameters S , and both of the constraints are also easily verified to be convex.
Sentence 2: the optimization problem is convex, which enables us to derive efficient, local-minima-free algorithms to solve it.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Indeed, the first step is due to the spherical symmetry of the hash family, the second step follows from the above discussion about replacing a random orthogonal matrix with a Gaussian one and that one can assume w.l.o.g.
Sentence 2: that p = e 1 and q = αe 1 + βe 2 ; the last step is due to the independence of the entries of u and v. proving Theorem 1 reduces to estimating the right-hand side of (1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For instance, for k = 0, the sub-importances of X 2 and X 5 are positive, while all others are null, which means that only those two variables are ever selected at the root node, hence masking the others.
Sentence 2: this also means that the importances of the remaining variables is biased and that it actually only accounts of their relevance when conditioned to X 2 or X 5 , but not of their relevance in other contexts.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This can be explained by Theorem 2 because the local incoherence of a block diagonal entry is given by µ ij = n 2 rK 2 , where K is the corresponding cluster size, and hence the error corruption probability should satisfy 1 2⇢ ij > C 0 p n K log n for correct recovery.
Sentence 2: a larger cluster can resist denser errors.
Relationship: reasoning