A Quadratic Lower Bound for Rocchio’s Similarity-Based Relevance Feedback Algorithm

Medical image retrieval can assist physicians in finding information supporting their diagnosis and fulfilling information needs. Systems that allow searching for medical images need to provide tools for quick and easy navigation and query refinement as the time available for information search is often short. Relevance feedback is a powerful tool in information retrieval. This study evaluates relevance feedback techniques with regard to the content they use. A novel relevance feedback technique that uses both text and visual information of the results is proposed. The two information modalities from the image examples are fused either at the feature level using the Rocchio algorithm or at the query list fusion step using a common late fusion rule. Results using the ImageCLEF 2012 benchmark database for medical image retrieval show the potential of relevance feedback techniques in medical image retrieval. The mean average precision (mAP) is used as the evaluation metric and the proposed method outperforms commonly-used methods. The baseline without feedback reached 16 % whereas the relevance feedback with 20 images reached up to 26.35 % with three steps and when using 100 images up to 34.87 % in four steps. Most improvements occur in the first two steps of relevance feedback and then results start to become relatively flat. This might also be due to only using positive feedback as negative feeback often also improves results after more steps. The effect of relevance feedback in automatically spelling corrected and translated queries is investigated as well. Results without mistakes were better than spell-corrected results but the spelling correction more than double results over non-corrected retrieval. Multimodal relevance feedback has shown to be able to help visual medical information retrieval. Next steps include integrating semantics into relevance feedback techniques to benefit from the structured knowledge of ontologies and experimenting on the fusion of text and visual information.

The effect of low-level image features on pseudo relevance feedback

Integrating visual word list and Rocchio algorithm, constructs an implicit relevance feedback retrieval model in semantic space, and judges information retrieval preference. User demand mining is introduced, Jensen-Shannon divergence is used to calculate the relative entropy distance between the probability distributions of document sets, and similarity matching is calculated to complete interactive information retrieval. Teaching ability is gradually formed by teachers in long-term teaching practice, and is stably and comprehensively reflected in teaching viewpoints, teaching methods, teaching skills, teaching style and so on. The weight analysis of each index in the evaluation index system was carried out by using the analytic hierarchy process, and the selection of the evaluation subject was completed through the third round of expert questionnaires, and the evaluation criteria were formulated in combination with the opinions of tutors and some expert teachers.

Traditional information retrieval algorithms ignore user needs and are unable to obtain user gaze coordinates and gaze times, resulting in low retrieval accuracy. The author proposes a new interactive information retrieval algorithm for this purpose. Divide eye tracking technology evaluation indicators, visually process eye movement information, obtain user gaze coordinates and gaze time, and calculate the influence coefficients of each gaze area and each point in the area. Weighted visual words are accumulated to get a visual word list with the weight of the associated area, and visual word list and Rocchio algorithm are combined to build a hidden Relevance feedback retrieval model in semantic space to judge information retrieval preferences. Intelligent robot is introduced, and Jensen Shannon divergence is used to calculate the Kullback–Leibler divergence distance between the probability distributions of document sets, calculate the similarity matching, and complete the interactive information retrieval. The simulation results prove that, due to the introduction of intelligent robot strategy in this method, the amount of irrelevant retrieval information is reduced by matching user needs and retrieval results. Therefore, the amount of messages generated by information retrieval is significantly lower than that of the two literature methods, without adding additional network load. The network system is in a stable operation state, and users can quickly grasp their own required information, and the retrieval speed is also improved to a certain extent. It is proved that the proposed algorithm has high retrieval accuracy, can effectively reduce network load and achieve high-quality human-computer interactive information retrieval.

Rocchio’s similarity-based Relevance feedback algorithm, one of the most important query reformation methods in information retrieval, is essentially an adaptive supervised learning algorithm from examples. In spite of its popularity in various applications there is little rigorous analysis of its learning complexity in literature. In this paper we show that in the Boolean vector space model, if the initial query vector is 0, then for any of the four typical similarities (inner product, dice coefficient, cosine coeffcient, and Jaccard coeffcient), Rocchio’s similarity- based relevance feedback algorithm makes at least n mistakes when used to search for a collection of documents represented by a monotone disjunction of at most k relevant features (or terms) over the n-dimensional Boolean vector space{0,1}n. When an arbitrary initial query vector in {0,1}n is used, it makes at least (n+k −3)/2 mistakes to search for the same collection of documents. The linear lower bounds are independent of the choices of the threshold and coefficients that the algorithm may use in updating its query vector and making its classification.

In this paper, we prove for the first time that the learning complexity of Rocchio’s algorithm is O(d + d 2(log d + log n)) over the discretized vector space {0,...,n–1}d, when the inner product similarity measure is used. The upper bound on the learning complexity for searching for documents represented by a monotone linear classifier (q,0) over {0,...,n–1}d can be improved to O(d + 2k(n–1)(log d + log(n–1))), where k is the number of nonzero components in q. An Ω((d 2)log n) lower bound on the learning complexity is also obtained for Rocchio’s algorithm over {0,...,n–1}d. In practice, Rocchio’s algorithm often uses fixed query updating factors. When this is the case, the lower bound is strengthened to 2\(^{{\it \Omega}(d)}\) over the binary vector space {0,1}d. In general, if the query updating factors are bounded by O(n c) for some constant c≥ 0, an \({\it \Omega}(n^{d-1-c}/(n-1))\) lower bound is obtained over {0,...,n–1}d.KeywordsInformation RetrievalNonzero ComponentRelevance FeedbackVector Space ModelLearning ComplexityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Chapter 5 - Vector Spaces, Hilbert Spaces, and the L2 Space

An integral formula for multiple summing norms of operators

1. Let R be the additive group of all real numbers and t → Tt a representation of this group on an n-dimensional complex vector space X. This means that to each real number t a linear operator Tt : X → X is assigned in such a way that $${\text{T}}_{{\text{t + s}}} = {\text{T}}_{\text{t}} {\text{T}}_{\text{s}} , {\text{T}}_0 = {\text{I}}$$ (1) Since the set {Tt : t ∈ R} consists of operators which commute one to the other the irreducible representations are one dimensional. Hence we are lead to find all functions f: R → C (C is the set of all complex numbers) such that $${\text{f}}\left( {{\text{t}}\,{\text{ + }}\,{\text{s}}} \right) = {\text{f}}\left( {\text{t}} \right)\,\,{\text{f}}\,\,{\text{(s),}}\,\,\,\,\,{\text{f}}\,\,{\text{(0)}}\,{\text{ = }}\,\,{\text{1}}{\text{.}}$$ (2)

On maximal compact subsemigroups of the endomorphism semigroup of an n-dimensional complex vector space

In this paper, we study the existence of extremal solutions for impulsive delay fuzzy differential equations in n-dimensional fuzzy vector space. This is an extension of the result of Kwun et al. [2] to impulsive fuzzy differential equations with delay condition.

A (differentiable) vector bundleof rank nconsists of a total space E,a base M,and a projection π :E → M,where Eand Mare differentiable manifolds, πis differentiable, each “fiber” Ex:= π -1(x) for x Є M,carries the structure of an n-dimensional (real) vector space, and the following local triviality requirement is satisfied: For each x Є M,there exist a neighborhood Uand a diffeomorphism.KeywordsRiemannian ManifoldVector BundleTangent BundleSpin StructureIntegral CurfThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The family \(\mathcal {N}\) of n-variate normal distributions is parameterized by the cone of positive definite symmetric \(n\times n\)-matrices and the n-dimensional real vector space. Equipped with the Fisher information metric, \(\mathcal {N}\) becomes a Riemannian manifold. As such, it is diffeomorphic, but not isometric, to the Riemannian symmetric space \(\mathrm {Pos}_1(n+1,\mathbb {R})\) of unimodular positive definite symmetric \((n+1)\times (n+1)\)-matrices. As the computation of distances in the Fisher metric for \(n>1\) presents some difficulties, Lovrič et al. (J Multivar Anal 74:36–48, 2000) proposed to use the Killing metric on \(\mathrm {Pos}_1(n+1,\mathbb {R})\) as an alternative metric in which distances are easier to compute. In this work, we survey the geometric properties of the space \(\mathcal {N}\) and provide a quantitative analysis of the defect of certain geodesics for the Killing metric to be geodesics for the Fisher metric. We find that for these geodesics the use of the Killing metric as an approximation for the Fisher metric is indeed justified for long distances.

In the whole chapter, we denote by E an n-dimensional Euclidean vector space. By a cross-section (for short, a section) of a lattice A, we mean the intersection of A with some subspace F of E. In general, it is to be understood that this intersection is indeed a (full) lattice in F.KeywordsRelative LatticeHyperplane SectionIntegral LatticePerfect LatticeMinimal VectorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A. Convex Sets. We begin here with some basic definitions and elementary facts. Throughout this book we will work in an n-dimensional real vector space \(\mathcal{X}\). The choice of a basis in \(\mathcal{X}\) amounts to a linear isomorphism \(\mathcal{X}\mathop{\cong}\mathbb{R}^{n}\), but we will make this identification only in examples and some explicit computations.