PleIAs/common_corpus · Datasets at Hugging Face

title: PleIAs/commoncorpus · Datasets at Hugging Face
contenttype: article
publication: Huggingface
published: 2024-10-29T00:00:00
sourceurl: https://huggingface.co/datasets/PleIAs/commoncorpus

word_count: 14973

hal-03712933-eccv2022submission.txt_1 Hierarchical Average Precision Training for Pertinent Image Retrieval. ECCV 2022, Oct 2022, Tel-Aviv, Israel. ⟨hal-03712933v2⟩ HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. Hierarchical Average Precision Training for Pertinent Image Retrieval

Elias Ramzi1,2, Nicolas Audebert1, Nicolas Thome1,3, Clément Rambour1 and Xavier Bitot2 1 CEDRIC, Conservatoire National des Arts et Métiers, Paris, France {elias.ramzi,nicolas.audebert,nicolas.thome,clement.rambour}@cnam.fr 2 Coexya, Paris, France xavier.bitot@coexya.eu 3 Sorbonne Université, CNRS, ISIR, F-75005 Paris, France

Abstract. Image Retrieval is commonly evaluated with Average Precision (AP) or Recall@k. Yet, those metrics, are limited to binary labels and do not take into account errors’ severity. This paper introduces a new hierarchical AP training method for pertinent image retrieval (HAPPIER). HAPPIER is based on a new H-AP metric, which leverages a concept hierarchy to refine AP by integrating errors’ importance and better evaluate rankings. To train deep models with H-AP, we carefully study the problem’s structure and design a smooth lower bound surrogate combined with a clustering loss that ensures consistent ordering. Extensive experiments on 6 datasets show that HAPPIER significantly outperforms state-of-the-art methods for hierarchical retrieval, while being on par with the latest approaches when evaluating fine-grained ranking performances. Finally, we show that HAPPIER leads to better organization of the embedding space, and prevents most severe failure cases of non-hierarchical methods. Our code is publicly available at https://github.com/elias-ramzi/HAPPIER.

Keywords: Hierarchical Image Retrieval, Hierarchical Average Precision, Ranking 1

Introduction

Image Retrieval (IR) consists in ranking images with respect to a query by decreasing order of visual similarity. IR methods are commonly evaluated using Recall@k (R@k) or Average Precision (AP). Because those metrics are non-differentiable , a rich literature exists on finding adequate surrogate loss functions to optimize them with deep learning, with tuple-wise losses [40,46,57,54,55], proxy based losses [59,53,12,49] and direct AP optimization methods [6,43,34,45,2,42]. These metrics are only defined for binary (⊕/) labels, which we denote as fine-grained labels: an image is negative as soon as it is not strictly similar to the query. Binary metrics are by design unable to take into account the severity of the mistakes in a ranking. On Fig. 1, some negative instances are “less negative” than others, e.g. given the “Brown Bear” query, “Polar bear” is more relevant than 2 E. Ramzi et al.

HAPPIER rank 1

rank 2 rank 3 rank 4 rank 5 rank

6 Query image. Baseline Fig. 1: Proposed HAPPIER framework for pertinent image retrieval. Standard ranking metrics based on binary labels, e.g. Average Precision (AP), assign the same score to the bottom and top row rankings (0.9). We introduce the H-AP metric based on non-binary labels, that takes into account mistakes’ severity. H-AP assigns a smaller score to the bottom row (0.68) than the top one (0.94). HAPPIER maximizes H-AP during training and thus

explicitly supports to learn rankings similar to the top one, in contrast to binary ranking

loss

es

.

“Butterfly”.

However

,

AP

is

0.9

for

both the top and

bottom rankings. Consequently, training on binary metrics (e.g. AP or R@k) develops no incentive to produce ranking such as the top row, and often produces rankings similar to the bottom one. To address this problem, we introduce the HAPPIER method dedicated to Hierarchical Average Precision training for Pertinent ImagE Retrieval. HAPPIER provides a smooth training objective, amenable to gradient descent, which explicitly takes into account the severity of mistakes when evaluating rankings. Our first contribution is to define a new Hierarchical AP metric (H-AP) that leverages the hierarchical tree between concepts and enables a fine weighting between errors in rankings. As shown in Fig. 1, H-AP assigns a larger score (0.94) to the top ranking than to the bottom one (0.68). We show that H-AP provides a consistent generalization of AP for the non-binary setting. We also introduce our HAPPIERF variant, giving more weights to fine-grained levels of the hierarchy. Since H-AP, like AP, is a non-differentiable metric, our second contribution is to use HAPPIER to directly optimize H-AP by gradient descent. We carefully design a smooth surrogate loss for H-AP that has strong theoretical guarantees and is an upper bound of the true loss. We then define an additional clustering loss to support having a consistency between partial and global rankings. We validate HAPPIER on six IR datasets, including three standard datasets (Stanford Online Products [33] and iNaturalist-base/full [51]), and three recent hierarchical datasets (DyML [47]). We show that, when evaluating on hierarchical metrics (e.g. H-AP), HAPPIER outperforms state-of-the-art methods for finegrained ranking [57,59,49,42], the baselines and the latest hierarchical method of [47], and only slightly under-performs vs. state-of-the-art IR methods at the fine-grained level (e.g. AP, R@1). HAPPIERF performs on par on fine-grained metrics while still outperforming fine-grained methods on hierarchical metrics. HAPPIER 2 Related work 2.1 Image Retrieval and ranking 3

The Image Retrieval community has designed several families of methods to optimize metrics such as AP and R@k. Methods that relies on tuplet-wise losses, like pair losses [17,41], triplet losses [57], or larger tuplets [46,27,54] learn comparison relations between instances. Methods using proxies have been introduced to lower the computational complexity of tuplet based training [31,59,53,12,49]: they learn jointly a deep model and weight matrix that represent proxies using a crossentropy based loss. Proxies are approximations of the original data points that should belong to their neighbourhood. Finally, there also has been large amounts of work dedicated to the direct optimization of the AP during training by introducing differentiable surrogates [6,43,34,45,2,42], so that models are optimized on the same metric they are evaluated on. However, nearly all of these methods only consider binary labels: two instances are either the same (positive) or different (negative), leading to poor performance when multiple levels of hierarchy are considered.

2.2 Hierarchical predictions and metrics

There has been a recent regain of interest in Hierarchical Classification [13,1,8] with the introductions of methods based either on a hierarchical softmax function or on multiple classifiers. It is considered that learning from hierarchical relations between labels leads to more robust models that make “better mistakes” [1]. Yet, hierarchical classification means that labels are known in advance and are identical in the train and test sets. This is called a closed set setting. However, Hierarchical Image Retrieval does not fall into this framework. Standard IR protocols consider the open set paradigm to better evaluate the generalization abilities of learned models: the retrieval task at test time pertains to labels that were not present in the train set, making classification poorly suited to IR. Meanwhile, the broader Information Retrieval community has been using datasets where documents can be more or less relevant depending on the query and the user making the request [20,24]. Instead of the mere positive/negative dichotomy, each instance has a continuous score quantifying its relevance to the query. To quantify the quality of their retrieval engine, Information Retrieval researchers have long used ranking based metrics, such as the NDCG [22,10], that penalize mistakes differently based on whether they occur at the top or the bottom of the ranking and whether wrong documents still have some marginal relevance or not. Average Precision is also used as a retrieval metric [23] and has even been given probabilistic interpretations based on how users interact with the system [14]. Several works have investigated how to optimize those metrics during the training of neural networks, e.g. using pairwise losses [4] and later using smooth surrogates of the NDCG in LambdaRank [5], SoftRank [48], ApproxNDCG [38] and LearningTo-Rank [3]. These works however focused on NDCG, the most popular metric for information retrieval, and are without any theoretical guarantees: the surrogates 4 E. Ramzi et al. are approximations of the NDCG but not lower bounds, i.e. their maximization does not imply improved performances during inference. An additional drawback of this literature is that NDCG does not relate easily to average precision [15], which is the most common metric in image retrieval. Fortunately, there have been some works done to extend AP in a graded setting where relevance between instances is not binary [44,14]. The graded Average Precision from [44] is the closest to our goal as it leverages SoftRank for direct optimization on non-binary relevance judgements, although there are significant shortcomings. There is no guarantee that the SoftRank surrogate actually minimizes the graded AP, it requires to annotate datasets with pairwise relevances which is unpractical for large scale settings and was only applied to small-scale corpora of a few thousands documents, compared to the hundred thousands of images in IR. Recently, the authors of [47] introduced three new hierarchical benchmarks datasets for image retrieval, in addition to a novel hierarchical loss CSL. CSL extends proxy-based triplet losses to the hierarchical setting and tries to structure the embedding space in a hierarchical manner. However, this method faces the same limitation as the usual triplet losses: minimizing CSL does not explicitly optimize a well-behaved hierarchical evaluation metric, e.g. H-AP. We show experimentally that our method HAPPIER significantly outperforms CSL [47] both on hierarchical metrics and AP-level evaluations.

3 HAPPIER Model

We detail HAPPIER our Hierarchical Average Precision training method for Pertinent ImagE Retrieval. We first introduce the Hierarchical Average Precision, H-AP in Sec. 3.1, that leverages a hierarchical tree (Fig. 2a) of labels. It is based on the hierarchical rank, H-rank, and evaluates rankings so that more relevant instances are ranked before less relevant ones (Fig. 2b). We then show how to directly optimize H-AP by stochastic gradient descent (SGD) using HAPPIER in Sec. 3.2. Our training objective combines a carefully designed smooth upper bound surrogate loss for LH-AP = 1 − H-AP and a clustering loss Lclust. that supports consistent rankings.

Context

Let us consider a retrieval set Ω = {xj }j∈J1;N K composed of N instances. For a query1 q ∈ Ω, we aim to order all xj ∈ Ω so that more relevant (i.e. similar) instances are ranked before relevant instances. In our hierarchical setting, the relevance of an instance xj is non-binary. We assume that we have access to a hierarchical tree defining semantic similarities between concepts as in Fig. 2a. For a query q, we leverage this knowledge to partition the set of retrieved instances into L+1 disjoint subsets Ω (l) l∈J0;LK. Ω (L) is the subset of the most similar instances to the query (i.e. fine-grained level): for L = 3 and a “Lada #2” query, Ω (3) are the images of the same “Lada #2” (green), see Fig. 2. The set Ω (l) for l < L contains instances with smaller 1 For the sake of readability, our notations are given for a single query. During training, HAPPIER optimizes our hierarchical retrieval objective by averaging several queries.

HAPPIER 5

Query Image: Lada #2 Vehicles Cars Lada Pickup Mini Bus Prius Prius #4 Lada #1 Lada #2 Lada #9 Fig. 2: HAPPIER leverages a hierarchical tree representing the semantic similarities between concepts in (a) to introduce a new hierarchical metric, H-AP in Eq. (3), see (b). H-AP exploits the hierarchy to weight rankings’ inversion: given the query image of a “Lada #2”, H-AP penalizes an inversion with a “Lada #9” less than with a “Prius #4”. To directly train models with H-AP, we carefully study the structure of the problem

and

introduce the LsH-AP loss

in

Eq. (5)

, which provides

a

smooth upper bound

of LH-AP

,

see (c).

We also

train

HAPPIER

with the

Lc

lust. loss

in Eq. (6)

to enforce the partial

ordering

in s

toch

astic

optimization to mach the global ones. relevance with respect to the query: Ω (2) in Fig. 2 is the set of “Lada” that are not “Lada #2” (blue) and Ω (1) is the set of “Cars” that are not “Lada” (orange). We also define Ω − := Ω (0) as the set of negativeSinstances, i.e. the set of vehicles L that are not “Cars” (in red) in Fig. 2 and Ω + = l=1 Ω (l). Each instance k of Ω (l) is thus associated a value through the relevance function denoted as rel(k) [20]. To rank the instances xj ∈ Ω with respect to the query q, we compute cosine similarities in an embedding space. More precisely, we extract embedding vectors using a deep neural network f parameterized by θ, vj = fθ (xj ), and compute the cosine similarity between the query and every image sj = fθ (q)T vj. Images are then ranked by decreasing cosine similarity score. We learn the parameters θ of the network with HAPPIER, our framework to directly minimize LH-AP (θ) = 1−H-AP(θ). This enforces a ranking where the instances with the highest cosine similarity scores belong to Ω (L), then Ω (L−1) etc. and the items with the lowest cosine similarity belong to Ω −.

3.1 Hierarchical Average Precision

Average Precision (AP) is the most common metric in Image Retrieval. AP evaluates a ranking in a binary setting: for a given query, each instance is either Query

Image: Lada #2

Fig. 3: Given a “Lada #2” query, the top inversion is less severe than the bottom one. Indeed on the top row instance 1 is semantically closer to the query – as it is a “Lada”– than instance 3 on the bottom row. Indeed instance 3’s closest common ancestor with the query, “Cars”, is farther in the hierarchical tree (see Fig. 2a). Because of that H-rank(2) is greater on the top row (5/3) than on the bottom row (4/3), leading to a greater H-AP in Fig. 2b for the top row. positive or negative. It is computed PN as the average of precision at each rank n over the positive set AP = |Ω1+ | n=1 Prec(n). Previous works have written the AP using the ranking operator [2] as in Eq. (1). The rank for an instance k is written as a sum of Heaviside (step) function H [38]: this counts the number of instances j ranked before k, i.e. that have a higher cosine similarity (sj > sk ). rank+ is the rank among the positive instances, i.e. restricted to Ω +. \label {eq:apdefinition} \text {AP} = \frac {1}{|\Omega ^+|} \sum {k\in \Omega ^+} \frac {\rank ^+(k)}{\rank (k)}, \; \text {with } \begin {cases} \rank (k) = 1 + \sum {j\in \Omega } H(sj - sk) \ \rank ^+(k) = 1 + \sum {j\in \Omega ^+} H(sj - sk) \end {cases} (1)

Extend

ing AP to hierarchical image retrieval

We propose an extension of AP that leverages non-binary labels. To do so, we extend the concept of rank+ to the hierarchical case with the concept of hierarchical rank, H-rank: \hrank (k) = \rel (k) + \sum {j\in \Omega ^+} \min (\rel (k), \rel (j))\cdot H(sj-sk) ~. \label {eq:hierarchicalrank} (2) Intuitively, min(rel(k), rel(j)) corresponds to seeking the closest ancestor shared by instance k and j with the query in the hierarchical tree. As illustrated in Fig. 3, H-rank induces a smoother penalization for instances that do not share the same fine-grained label as the query but still share some coarser semantics, which is not the case for rank+. From H-rank in Eq. (2) we define the Hierarchical Average Precision, H-AP: \label {eq:def_hap} \hap = \frac {1}{\sum _{k\in \Omega ^+}\rel (k)} \sum _{k\in \Omega ^+} \frac {\hrank (k)}{\rank (k)} (3) Eq. (3) extends the AP to non-binary labels. We replace rank+ Pby our hierarchical rank H-rank and the normalization term |Ω + | replaced by k∈Ω + rel(k), which both represent the “sum of positives”, see more details in supplementary A.2. HAPPIER 7

H-AP extends the desirable properties of the AP. It evaluates the quality of a ranking by: i) penalizing inversions of instances that are not ranked in decreasing order of relevances with respect to the query, ii) giving stronger emphasis to inversions that occur at the top of the ranking. Finally, we can observe that, by this definition, H-AP is equal to the AP in the binary setting (L = 1). This makes H-AP a consistent generalization of AP (details in supplementary A.2).

Relevance function design

The relevance rel(k) defines how “similar” an instance k ∈ Ω (l) is to the query q. While rel(k) might be given as input in Information Retrieval datasets [37,9], we need to define it based on the hierarchical tree in our case. We want to enforce the constraint that the relevance decreases ′ when going up the tree, i.e. rel(k) > rel(k ′ ) for k ∈ Ω (l), k ′ ∈ Ω (l ) and l > l′. To do so, we assign a total weight of (l/L)α to each semantic level l, where α ∈ R+ controls the decrease rate of similarity in the tree. For example for L = 3 and α = 1, the total weights for each level are 1, 23, 13 and 0. The instance relevance rel(k) is normalized by the cardinal of Ω (l) : \label

{

eq

:hier

archy

_relevance} \rel (k) = \frac {(l/L)^\alpha }{|\Omega ^{(l)}|} \; \text {if } k \in \Omega ^{(l)} (4)

Other definitions fulfilling the decreasing similarity behaviour in the tree are possible. An interestingP option forP the relevance enables to recover a weighted L sum of AP, denoted as wAP := l=1 wl ·AP(l) ( supplementary A.2), i.e. the weighted sum of AP is a particular case of H-AP. We set α = 1 in Eq. (4) for the H-AP metric and in our main experiments. Setting α to larger values supports better performances on fine-grained levels as their relevances will relatively increase. This variant is denoted HAPPIERF and discussed in Sec. 4. 3.2

Direct optimization of H-AP H-AP

in

Eq. (3)

involv

es the computation of H-rank

and

rank

, which are nondifferentiable due to the summing of

Heaviside

step function

s

. We thus introduce a smooth approximation of H-AP to obtain a surrogate loss amenable to gradient descent, which fulfils theoretical guarantees for proper optimization

. Re-writing H-AP

In order to design our surrogate loss for LH-AP = 1−H-AP, we decompose H-rank and rank into two quantities. Denoting H-rank> (k) (resp. H-rank≤ (k)) as the restriction of H-rank to instances of strictly higher relevances (resp. lower or equal), we can see that H-rank(k) = H-rank> (k) + H-rank≤ (k). The rank can be decomposed in a similar fashion: rank(k) = rank≥ (k)+rank< (k) where < (resp. ≥) denotes the restriction to instances of strictly lower relevances (resp. higher or equal). The LH-AP can be rewritten as follow: \label {

eq:re

write

_hap} \lhap = 1 - \frac {1}{\sum _{k\in \Omega ^+}\rel (k)} \sum _{k\in \Omega ^+} \frac {\hrank ^>(k) + \hrank ^\leq (k)}{\rank ^\geq (k) + \rank ^<(k)}~. (5) 8 E. Ramzi et al. We choose to optimize over H-rank> and rank< in Eq. (5). We maximize H-rank> to enforce that the k th instance must decrease in cosine similarity score if it is ranked before another instance of higher relevance (∇H-rank> in Fig. 2 enforces the blue instance to be ranked after the green one as it is less relevant to the query). We minimize rank< to encourage the k th instance to increase in cosine similarity score if it is ranked after one or more instances of lower relevance (∇rank< in Fig. 2 enforces that the last green instance moves before less relevant instances). Optimizing both those terms leads to a decrease in LH-AP. On the other hand, we purposely do not optimize the two remaining H-rank≤ (k) and rank≥ (k) terms, since this could harm training performances as explained in supplementary A.3.

Upper bound of LH-AP

Based on the previous analysis, we now design our surrogate loss LsH-AP by introducing a smooth approximation of rank< and H-rank> (k). An important sought property of LsH-AP is that it is an upper bound of LH-AP. To this end, we approximate H-rank> (k) with a piece-wise linear function that is a lower bound of the Heaviside function. rank< is approximated with a smooth upper bound of the Heaviside that combines a piece-wise sigmoid function and an affine function, which has been shown to make the training more robust thanks to the induced implicit margins between positives and negatives [45,2,42]. More details are given in supplementary A.3 on those surrogates. Clustering constraint in HAPPIER

Positives only need to have a greater cosine similarity with the query than negatives in order to be correctly ranked. Yet, we cannot optimize the ranking on the entire datasets – and thus the true LH-AP – because of the batch-wise estimation performed in stochastic gradient descent. To mitigate this issue, we take inspiration from clustering methods [59,49] to define the following objective in order to group closely the embeddings of instances that share the same fine-grained label: \label {eq:clusterloss} \lclust (\theta ) = - \log \left ( \frac {\exp (\frac {vy^T py}{\sigma })}{\sum {{pz}\in \mathcal {Z}} \exp (\frac {vy^T pz}{\sigma })} \right ), (6) where py is the normalized proxy corresponding to the fine-grained class of the embedding vy, Z is the set of proxies, and σ is a temperature scaling parameter. In Fig. 2, ∇Lclust. further clusters “Lada #2” instances. Lclust. induces a reference shared across batches and thus enforces that the partial ordering in-between batches is consistent with the global ordering over the entire retrieval set. Our resulting final objective is a linear combination of both our losses, with a weight factor λ ∈ [0,1] that balances the two terms: \mathcal {L}{\text {HAPPIER}}(\theta ) = (1-\lambda )\cdot \lhaps (\theta ) + \lambda \cdot \lclust (\theta ) ~. HAPPIER 4 Experiments 4.1 Experimental setup 9 Datasets

We use the standard benchmark Stanford Online Products [33] (SOP) with two levels of hierarchy (L = 2), and iNaturalist-2018 [51] with the standard splits from [2] in two settings: i) iNat-base with two levels of hierarchy (L = 2) ii) iNat-full with the full biological taxonomy composed of 7 levels (L = 7). We also evaluate on the recent dynamic metric learning (DyML) datasets (DyML-V, DyML-A, DyML-P) introduced in [47] for the task of hierarchical image retrieval, each with 3 semantic levels (L = 3).

Implementation details

Our base model is a ResNet-50 pretrained on ImageNet for SOP and iNat-base/full, and a ResNet-34 randomly initialized on DyML-V&A and pretrained on ImageNet on DyML-P, following [47]. Unless specified otherwise, all reported results are obtained with α = 1 in Eq. (4) and λ = 0.1 for LHAPPIER. We study the impact of these parameters in Sec. 4.3. Metrics For SOP and iNat, we evaluate the models based on three hierarchical metrics: H-AP – which we introduced in Eq. (3) – the Average Set Intersection (ASI) and the Normalized Discounted Cumulative Gain (NDCG), defined in supplementary B.3. We also report the AP for each semantic level. For DyML, we follow the evaluation protocols of [47] and compute AP, ASI and R@1 on each semantic scale before averaging them. We cannot compute H-AP or NDCG on those datasets as the hierarchical tree is not available on the test set. Baselines

We compare HAPPIER to several recent image retrieval methods optimized at the fine-grained level, which represent strong baselines for IR when training with binary labels: Triplet SH (TLSH ) [57], NormSoftMax (NSM) [59], ProxyNCA++ (NCA++) [49] and ROADMAP [42]. We also benchmark against hierarchical methods obtained by summing these fine-grained losses at different levels (denoted by Σ), and with respect to the recent hierarchical CSL loss [47]. Details on the experimental setup are given in supplementary B.

4.2 Main Results Hierachical results

We first evaluate HAPPIER on global hierarchical metrics. On Tab. 1, we notice that HAPPIER significantly outperforms methods trained on the fine-grained level only, with a gain on H-AP over the best performing methods of +16.1pt on SOP, +13pt on iNat-base and 12.7pt on iNat-full. HAPPIER also exhibits significant gains compared to hierarchical methods. On H-AP, HAPPIER has important gains on all datasets (e.g. +6.3pt on SOP, +4.2pt on iNat-base over the best competitor), but also on ASI and NDCG. This shows the strong generalization of the method on standard metrics. Compared to the recent CSL loss [47], we observe a consistent gain over all metrics and datasets, e.g. +6pt on H-AP, +8pt on ASI and +2.6pts on NDCG on SOP. This shows the benefits of optimizing a well-behaved hierarchical metric compared to an ad-hoc proxy method.

10

E. Ramzi et al.

Table 1: Comparison of HAPPIER on SOP and iNat-base/full when using hierarchical metrics. Best results in bold, second best underlined. SOP Hier. Fine Method iNat-base iNat-full H-AP ASI NDCG H-AP ASI NDCG H-AP ASI NDCG Triplet SH [57] NSM [59] NCA++ [49] Smooth-AP [2] ROADMAP [42] 42.2 42.8 43.0 42.9 43.3 22.4 21.1 21.5 20.6 19.1 78.8 78.3 78.4 78.2 77.9 39.5 38.0 39.5 41.3 40.3 63.7 51.6 57.0 64.2 61.0 91.5 88.9 90.1 91.9 91.2 36.1 33.3 35.3 37.2 34.7 59.2 51.7 55.7 60.1 59.6 89.8 88.2 89.0 90.1 89.5 ΣTLSH [57] ΣNSM [59] ΣNCA++ [49] CSL [47] 53.1 50.4 49.5 52.8 53.3 49.7 52.8 57.9 89.2 87.0 87.8 88.1 44.0 47.9 48.9 50.1 87.4 75.8 78.7 89.3 96.4 94.4 95.0 96.7 39.9 46.9 44.7 45.1 85.5 74.2 74.3 84.9 92.0 93.8 92.6 93.0 HAPPIER 59.4 65.9 91.5 54.3 89.3 96.9 47.9 87.2 93.8

On Tab. 2, we evaluate HAPPIER on the recent DyML benchmarks. HAPPIER again shows significant gains in mAP and ASI compared to methods only trained on fine-grained labels, e.g. +9pt in mAP and +10pt in ASI on DyML-V. HAPPIER also outperforms other hierarchical baselines: +4.8pt mAP on DyMLV, +0.9 on DyML-A and +1.8 on DyML-P. In R@1, HAPPIER performs on par with other methods on DyML-V and outperforms other hierarchical baselines by a large margin on DyML-P: 63.7 vs. 60.8 for ΣNSM. Interestingly, HAPPIER also consistently outperforms CSL [47] on its own datasets2.

Table 2: Performance comparison on Dynamic Metric Learning benchmarks [47]. Hier. Fine Method 2 DyML-Vehicle DyML-Animal DyML-Product mAP ASI R@1 mAP ASI R@1 mAP ASI R@1 TLSH [57] NSM [59] Smooth-AP [2] ROADMAP [42] 26.1 27.7 27.1 27.1 38.6 40.3 39.5 39.6 84.0 88.7 83.8 84.5 37.5 38.8 37.7 34.4 46.3 48.4 45.4 42.6 66.3 69.6 63.6 62.8 36.32 35.6 36.1 34.6 46.1 46.0 45.5 44.6 59.6 57.4 55.0 62.5 ΣTLSH [57] ΣNSM

[

59] CSL [47] 25.5 32.0 30.0 38.1 45.7 43.6 81.0 89.4 87.1 38.9 42.6 40.8 47.2 50.6 46.3 65.9 70.0 60.9 36.9 36.8 31.1 46.3 46.9 40.7 58.5 60.8 52.7 HAPPIER 37.0 49.8 89.1 43.8 50.8 68.9 38.0 47.9 63.7

CSL’s score on Tab. 2 are above those reported in [47]; personal discussions with the authors [47] validate that our results are valid for CSL, see supplementary B.5. HAPPIER 11 Detailed evaluation Tabs. 3 and 4 shows the different methods’ performances on all semantic hierarchy levels. We evaluate HAPPIER and also HAPPIERF (α > 1 for Eq. (4) in Sec. 3.1), with α = 5 on SOP and α = 3 on iNat-base/full. HAPPIER optimizes the overall hierarchical performances, while HAPPIERF is meant to be optimal at the fine-grained level while still optimizing coarser levels.

Table 3:

Comparison of HAPPIER vs. methods trained only on fine-grained labels on SOP and iNat-base. Metrics are reported for both semantic levels. SOP Hier. Fine Fine iNat-base Method R@1 AP Coarse AP R@1 Fine AP Coarse AP TLSH [57] NSM [59] NCA++ [49] Smooth-AP [2] ROADMAP [42] 79.8 81.3 81.4 81.3 82.2 59.6 61.3 61.7 61.7 62.5 14.5 13.4 13.6 13.4 12.9 66.3 70.2 67.3 67.3 69.3 33.3 37.6 37.0 35.2 35.1 51.5 38.8 44.5 53.1 50.4 CSL [47] 79.4 58.0 45.0 62.9 30.2 88.5 HAPPIER HAPPIERF 81.0 81.8 60.4 62.2 58.4 36.0 70.7 71.0 36.7 37.8 88.6 78.8

On Tab. 3, we observe that HAPPIER gives the best performances at the coarse level, with a significant boost compared to fine-grained methods, e.g. +43.9pt AP compared to the best non-hierarchical TLSH [57] on SOP. HAPPIER even outperforms the best fine-grained methods in R@1 on iNat-base, but is slightly below on SOP. HAPPIERF performs on par with the best methods at the finest level on SOP, while further improving performances on iNat-base, and still significantly outperforms fine-grained methods at the coarse level. The satisfactory behaviour and the two optimal regimes of HAPPIER and HAPPIERF are confirmed and even more pronounced on iNat-full (Tab. 4): HAPPIER gives the best results on coarser levels (from “Order”), while being very close to the best results on finer ones. HAPPIERF gives the best results at the finest levels, even outperforming very competitive fine-grained baselines. Again, note that HAPPIER outperforms CSL [47] on all semantic levels and datasets on Tabs. 3 and 4, e.g. +5pt on the fine-grained AP (“Species”) and +3pt on the coarsest AP (“Kingdom”) on Tab. 4. 4.3 HAPPIER analysis Ablation study

In Tab. 5, we study the impact of our different choices regarding the direct optimization of H-AP. The baseline method uses a sigmoid to optimize H-AP as in [38,2]. Switching to our surrogate loss LsH-AP Sec. 3.2 yields a +0.8pt increase in H-AP. Finally, the combination with Lclust. in HAPPIER results in an additional 1.3pt improvement in H-AP.

12 E. Ramzi et al

. Table 4: Comparison of HAPPIER vs. methods trained only on fine-grained labels on iNat-Full. Metrics are reported for all 7 semantic levels. Genus Family Order Class R@1 Species AP AP AP AP AP AP AP TLSH [57] NSM [59] NCA++ [49] Smooth-AP [2] ROADMAP [42] 66.3 70.2 67.3 67.3 69.3 33.3 37.6 37.0 35.2 35.1 34.2 38.0 37.9 36.3 35.4 32.3 31.4 33.0 33.5 29.3 35.4 28.6 32.3 35.0 29.6 48.5 36.6 41.9 49.3 46.4 54.6 43.9 48.4 55.8 54.7 68.4 63.0 66.1 69.9 69.5 CSL [47] 59.9 30.4 32.4 36.2 50.7 81.0 87.4 91.3 HAPPIER HAPPIERF 70.2 70.8 36.0 37.6 37.0 38.2 38.0 38.8 51.9 50.9 81.3 76.1 89.1 82.2 94.4 83.1 Hier. Fine Method Phylum Kingdom Table 5: Impact of optimization choices Table 6: Comparison of H-AP (Eq. (4)) for H-AP (cf. Sec. 3.2) on iNat-base. and ΣwAP from supplementary A.2. LsH-AP Lclust. H-AP ✗ ✓ ✓ ✗ ✗ ✓ 52.3 53.1 54.3 test→ train↓ H-AP P wAP NDCG H-AP P wAP 53.1 52.0 39.8 40.5 97.0 96.4 Impact of

the relevan

ce function

Tab. 6 compares P models that are trained with the relevance function of Eq. (4), i.e. H-AP, andP wAP (relevance given in supplementary A.2). We report results for H-AP, wAP and NDCG. Both P H-AP, wAP perform better when trained withP their own metric: +1.1pt H-AP for the model trained to optimize it and +0.7pt wAP for the model trained to optimize it. Both models show similar performances in NDCG (96.4 vs. 97.0). 54 37.6 52

H-AP AP

fine 53 37.4 37.2 51 50 37.0 49

APfine

36.8

1.0 1.5 2.0 3.0 4.0 5.0 H-AP 48 0.0 0.1 0.2 0.3 0.4 0.5 0.7 α λ (a) APfine vs α in Eq. (4). (b) H-AP vs. λ for LHAPPIER. Fig. 4: Impact on Inat-base of α in Eq. (4) for setting the relevance of H-AP (a) and of the λ hyper-parameter on HAPPIER results (b).

HAPPIER 13 Hyper-parameters

Fig. 4a studies the impact of α for setting the relevance in Eq. (4): increasing α improves the performances of the AP at the fine-grained level on iNat-base, as expected. We also show in Fig. 4b the impact of λ weighting LsH-AP and Lclust. in HAPPIER performances: we observe a stable increase in H-AP within 0 < λ < 0.5 compared to optimizing only LsH-AP, while a drop in performance is observed for λ > 0.5. This shows the complementarity of LsH-AP and Lclust., and how, when combined, HAPPIER reaches its best performance.

4.4 Qualitative study

We provide here qualitative assessments of HAPPIER, including embedding space analysis and visualiz ation of HAPPIER’s retrievals. t-SNE: organization of the embedding space

In Fig. 5, we plot using tSNE [50,7] how HAPPIER learns an embedding space on SOP (L = 2) that is well-organized. We plot the mean vector of each fine-grained class and we assign the color based on the coarse level. We show on Fig. 5a the t-SNE visualisation obtained using a baseline method trained on the fine-grained labels, and in Fig. 5b we plot the t-SNE of the embedding space of a model trained with HAPPIER. We cannot observe any clear clusters for the coarse level on Fig. 5a, whereas we can appreciate the the quality of the hierarchical clusters formed on Fig. 5b. Controlled errors Finally, we showcase in Fig. 6 errors of HAPPIER vs. a fine-grained baseline. On Fig. 6a, we illustrate how a model trained with HAPPIER makes mistakes that are less severe than a baseline model trained only on the fine-grained level. On Fig. 6b, we show an example where both models fail to retrieve the correct fine-grained instances, however the model trained with HAPPIER retrieves images of bikes that are visually more similar to the query. (a) t-SNE visualization of a model (b) t-SNE visualization of a model trained only on the fine-grained labels. trained with HAPPIER.

Fig. 5: t-SNE visualisation of the embedding space of two models trained on SOP. Each point is the average embedding of each fine-grained label (object instance) and the colors represent coarse labels (object category, e.g. bike, coffee maker). rank 2 rank 3 rank 4 rank 5 rank 6 HAPPIER

rank 1

Baseline Query image

(a) HAPPIER can help make less severe mistakes. The inversion on the bottom row are with negative instances (in red), where as with HAPPIER (top row) inversions are with instances sharing the same coarse label “bike” (in orange).

rank 2 rank 3 rank 4 rank 5 rank 6 HAPPIER rank 1 Baseline Query image

(b) In this example, the models fail to retrieve the correct fine grained images. However HAPPIER still retrieves images of very similar bikes (in orange) whereas the baseline retrieves images that are dissimilar semantically to the query (in red).

Fig. 6: Qualitative examples of failure cases from a standard fine-grained model corrected by training with HAPPIER.

5 Conclusion

In this work, we introduce HAPPIER, a new training method that leverages hierarchical relations between concepts to learn robust rankings. HAPPIER is based on a new metric H-AP that evaluates hierarchical rankings and uses a combination of a smooth upper bound surrogate with theoretical guarantees and a clustering loss to directly optimize it. Extensive experiments show that HAPPIER performs on par to state-of-the-art image retrieval methods on fine-grained metrics and exhibits large improvements vs. recent hierarchical methods on hierarchical metrics. Learning more robust rankings reduces the severity of ranking errors, and is qualitatively related to a better organization of the embedding space with HAPPIER. Future works include the adaptation of HAPPIER to the unsupervised setting, e.g. for providing a relevant self-training criterion.

Acknowledgement HAPPIER 15

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Vig. n. 190; but ei@e, ef yép and ai ydép are more free. v. ef ydp.—6. with Optat., as a sort of particle of time, of repeated actions, as often as, whenever, Thuc. 7, 79, usu. with impf. or plqpf., sometimes with aor.—II. wirH INDIC., where possibility is asserted, without expressing any uncertainty or question; if, since.—1. with indic. pres., ef pw’ Eoger ToAEuilew, GAAovE wev ners or, Il. 3,67, where no doubt is thrown on the supposition.—2. with ‘indic. past, esp. in oaths and prayers, ef NOTE ToL éxi vndv Epewa, THdE [0b Kpynvov &éAdwp Il. 1, 39, etc., v. el- toTe.—3. with indic. fut NOTE ToL éxi vndv Epewa, THdE [0b Kpynvov &éAdwp Il. 1, 39, etc., v. el- toTe.—3. with indic. fut NOTE ToL éxi vndv Epewa, THdE [0b Kpynvov &éAdwp Il. 1, 39, etc., v. el- toTe.—3. with indic. fut NOTE ToL éxi vndv Epewa, THdE [0b Kpynvov &éAdwp Il. 1, 39, etc., v. el- toTe.—3. with indic. fut NOTE ToL éxi vndv Epewa, THdE [0b Kpynvov &éAdwp Il. 1, 39, etc., v. el- toTe.—3. with indic. fut NOTE ToL éxi vndv Epewa, THdE [0b Kpynvov &éAdwp Il. 1, 39 , yydceat, el Kai Beorecin TOALY ob GAaTaésetc, Il. 2, 367,379, where the fut. is looked on ascertain: Att. the optat: with av freq. follows, to soften the positiveness of the phrase, Soph. El. 244. So the indic. often follows, even after the opt. expressing a simple supposed case, e. g. weot pevoiveor, el TeAéovorr, Il. 12,59, they tried whether they could; where they are represented as it were saying, We will try whether we can, O as to add vivacity to the sentence: esp. oft. in Att. Prose. The indic. pres. or fut. is also put after ef in protasis, when not a mere probability, but a necessary result on a condition is intended, Il. 5, In Att., e¢ with indic. is used not only of probable, but of actual events, to qualify the positive assertion, and so much like 671: most freq. after Gavudlo, also after other verbs, esp. expressing strong feeling, e. g. dyavaxtéw, detvov ToLoduat, dndoi, ete., Hdt. 1, 155, Thuc. 6, 60, Plat. Lach. 194 A.—4. In Att. where ei with impf. is followed by dy with impf., the first implies that a condition has not been fulfilled, the second that a result has therefore not taken place; e. g. ef Te elyev, édidov dy, if he had it, he would give it... (but he has it not.)—5. with indic. aor., followed by indic. aor. with dv, it expresses the same thing in reference to a past time, for which in Lat. both verbs would have been in subj. plapf., ef rz dy, Edwxev dv had he had it, he would have given it, cf. Il. 21, 211, 544. In this case the impf. with dv may follow, ei érreicOnv, ok dv Hppdcrovr, had 1 obeyed, I should not have been ill, Buttm. Gramm. § 139, 9, 4. and 10: sometimes, but not often, this day is left out with the impf. 7v, Thuc. 1, 37. More rarely the opt. with day follows ei wf and the indic. aor., Il. 5, 388 ;17, 70.—I. wirH suBsUNCT., ei is scarcely to be distinguished from édv, though an attempt has been made to explain ef as expressing greater probability in the condition, suppose that, Kihner Ausf. Gr. § 818, Anm. 1, Herm. Soph. Ant. 706 ; much more rare than the former, but most freq. in Hom., Il. 1, 340, Od. 5, 221, etc.: ef kev with subjunct. being the more freq. For the Att. it was for- merly laid down that only éév or 77v, never ef was used with subjunct. : but many exceptions are found in Trag., as Soph. O. T. 198, 874, O. C. 1443, Ant. 710, 1032, cf. Herm. Aj. 491: also in comic wr., as Ar. Eq. 698, 700, Pac. 450.: nay it has been admitted even in prose, as Thuc. 6, 21, Xen. Mem. 2,.1, 12, Plat. Phaedr. 234, Rep. 579 E: in later‘authors «/ veth subjunct. is very common, with cptat., without apodosis, # «- | BH rm. Vig. n. 304: cf. also ef Ke— EIAP IV. with ParTicip. instead of in fie where oti is usu. supplied, but rave Soph. Aj. 686, and Herm. ib. 179, Bornem. Xen. Mem. 2, 6, 25.—V. WITH INFIN., sometimes in Hat. e g. 3, 105, 108, in orat. obliqua.— Ey from the first clause must sometimes be supplied with each of several fol- lowing clauses, even when these are in different moods, Schaf. Mel. p. 111. Whether, in indirect questions and after verbs containing a question, doubt, uncertainty, etc., or Bede, etc., we do not know whether to be a god, Il. 5, 183; in Hom. also free in ellipt. clauses, where wepoevoc, okorov, etc., must be supplied, etc. KnpdKecot KéAEvoaY, Gui TPL OTi out Tpinoda péyay, (TELPNOGLEVvOL) & memiOotey Ilndeidnv, trying whether they could move Achilles, Il. 23, 40; where the optat. without dv is used, because the action is past; ef. fl. 10, 206; 20, 464; if present or future, it would require e¢ xe or ééy with subj., Il. 5, 279, though Att. ef with subj. is used even in this sign. — C. Regularly ei begins the sentence, and so is followed by the articles: hence all compds., as & Keé, el mp, el uh, et Kal, etc., may, be best referred to their own specific heads. It is preceded by one or two conjunctions: — I. «ai el and if, even though, implying that the case is not so, Il. 20, 371; Kai et mov, Od. 7, 320, also kai ef ke, which follows the same rules as ef ce, Att. cel, xaév, Kavei: in Att. also 6uwe is oft. added in ap odosis (even though, yet still), though this word is sometimes attached to the end of the conditional clause, to which it adds force, Aesch. Pers. 295, Cho. 115: care must be taken not to confuse the end with ed, Herm. Vig. n. 307.—II. ob? ei, nay not if, not even if, Il. 5, 645; 20, 102, Od. 4, 293. —III. ed. and ei Te or (as Wolf writes it) cel, ooel Te, as if, as though, in comparisons, Od. 7, 36, I, 13, 492, 19, 366, Od. 19, 39: the Att. also inserts dy or wep, Gorep él, O¢ av el, Gorep dv ei or Worepavel, Heind. Plat. Gorg. 479 A. Ei, Dor. for 7 and of, cf. mez. Eid, also properisp. ela, and poet. trisyll. éia, Lat. eia, a cheering or stimulating exclamation, on! up! away! Trag., etc.: also come on then! Aesch. Ag. 1650, and Plat. ; ela vuv, well now! Ar. Pac. 459, stronger than dye vuy: also ela Oy: éa and etaare akin to it. [@ al- ways, whence Gramm. wrote éia, Vv. Reisig de Constr. Antistr. p. 19.] Eig, 3 sing. imperf. act. from édo, Hom. ; Eidew, f. -dow, to cry sia, like aidGw from ai, and ebdfw from eda, v, Valck. Diatr. p. 20. Eiduevy, ie, 7. & low, moist pasture, water-meadow, év eiaweva EAeoc, Il. 4. 483, in Ap. Rh. a flooded meadow. (Usu. deriv. from efatat, jvTat, neat, juevoc, Whence some Gramm. wrote ciapevy, cf. xéOnuwac: Buttm, however, V. #ldecc, connects it with Hiov.) ; Ei dv, Ep. and Ion. e ke Contr. into ééy and jy. But et...dv seems permissible both in Hom., and Att., where some words come between, Il. 2, 597, cf Herm. Vig. n. 303, Elavoc, 7, 6v, Ep. for éaéc, Il. 16,9. Elapéuacboc, ov, (slap, pacbdc\ with youthful, sweetly breasts, Ant. 397 Eldpo- span, &c, (etap, TépToual) joying in spring, Orph. Eldpo- span, &c, (etap, TépToual) joying in spring, Hom., Elatat, eiaro, 3 pl. pres. and impf., poet. for Ion. Earaz, gato, and this for qvraz, WTO, from 7juat, Hom. filato, 3 plur. imperf. mid. from et‘, for #vTo, i.e. #oav, occurs only Ou. 20, 106, where Buttm. Ausf. Gr. Anm. 14, n., would read ei-aro. Efaro, 3 sing. plqpf. mid. from év- pout for elvto, they had on. E/Bipoc, ov, trickling: from EI’BQ, Ep. form of AeiBw, to drop, let fall in drops, Hom., who regul. uses it in phrase daxpvoy eiBecy and Kata Odxpvov elBewv, to shed tears. Mid. to trickle or run down, drip, Hes. ‘Th. 210: but also as in act., ddxpva elBouévn, Soph. Ant. 527. Ei ydp, for if..., Il. 20, 26: but usu. —II. expressing a wish, O 7f..., O that..., would that... Lat. utimam! c. optat., ed yap ’A@nvn doin Kapto¢ éuoé, Il. 17, 561, so ef yap tot, Od. 17, 513, and ef yap mwc, Od. 16, 148. But Hom. ig tage has ai ydp, ai vip On, at yap 6n ToTE, at yap Two. Tike (Moraine: use c. inf. is rare, at yap, Too &WV;...240¢ yauBpo¢g Ka- A€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€ wish without alluding to its result, Nitzsch Od. 1, 265. Elye, if however, adding a condition which makes the thing dependent upon it unlikely or impossible, e.g., oikévee e0éAetc (évat etye wév el- detc, 6ooa Tol aica Kyde’ dvaTAn- out, évOade k’ ade wév tode OGua ov 4ococc, thou wishest to go home: yet of thou didst know..., ete., Od. 5, 206,—II.. if then, since, Lat. siquidem, of things which are taken for granted, Il. 1, 393, Od. 16, 300, Herm. Vig. n. 310. Cf. etzep. Ei yodv, even if, implying that the thing will not be so, only once Hom., viz. Il. 5; 258, ubi al. ef y ody. Ei 0’, dye, used in cheering, etc., on then, come on! oft. in Hom., who also has plur., e/ 0’, dyere, Il. 22, 381. He usu. joins ei. 0’, dye viv, el 0’; aye On, et. 0’, dye unv, or el 0, dye tot, followed by imperat., also ef 0’, tye Tol, Od. 9, 37. For the imperat. dedpo is, found, Il. 17, 685, and in speaking to one’s self the subj. aor., Od. 9, 37, or indic. fut., Il. 1, 524; 9, 167. The phrase is elliptic, and would be in full ef &’ é0¢Aece or ef dé BobAet, kye, but if thou wishest, come, and so serves to qualify the imperat., like Lat. sis vide, fac sis, agite sultis, Nitzsch Od. 1, 270. Eidaévouat, poet. lengthd. form of eldomat, to be like, rwvi, Nic. ElddAmoc, n, ov, (eidog) formed; here.ce shapely, comely, Od. 24, 279.— [L. like, looking like, Anth. ElddAAouat, = eldaivouat, ivdda- AQUAL. ElddAAouat, = eldaivouat, ivdda- AQUAL. ElddAAouat, arog, 76, (&du, as if lengthd. poet. from &dap) food, meat, victuals, Hom.—2 Of cattle, fodder, forage, etc. 5, 369.—3. also a bait for fish, Od. 12, 252.—4. weAicone dvOiuov eidap, of honey-cakes, Theocr. 15, 115. Ep. word, Ei 0é, with no apodosis, is elliptic, as Il. 9, 46, ef dé kal adrol, devydv-ran, but a eer (will), let them flee, where é0éAovcr is to be supplied, as in ef 0’, dye: so too 9, 262, e dé, (0€Aetc), a ee pev Gxovoov. In Il. 21, 487, and Od. 2, 115, the apodosis is implied in the protasis.—II. in complete sentences, but if, even if, oft. in Hom. It may be followed by any particle which follows e/, v. esp. El wéy: on ei 0’ od and ef 0’ ovv V. ec un. We have the notion of et dé strengthd. in ed 0’ av, if on the other hand, Od. 16, 105. Eidéa, ac, 7, for idéa, dub. in Ar. Thesm. 438. Eidéyera, opt., and eidévaz, inf. of olda, q. V. Ei 02 pH, v. sub et 7}. Eidéyera, ac, #, an odious, ugly look, LX X.: from EldéyOae, é¢, (eldoc, &yBoc) of hateful look, in genl. ugly, Polyb.: putrid, fetid, Hipp. Eidéw, for eid, subj. from oida. Ki 67, expressing a supposition which cannot be contradicted, 7f now, seeing that, ll. 1, 61, esp. after 7, Il. 1, 294, 574: also in Indirect questions, whether now, Od. 1, 207: always c. indicat. Eidjua, aroc, 76, (eidévar) knowledge. Eidnovixdc, ady., with knowledge, skilfully. Eidnovix, ov, gen. ovoc, (eldévat) knowing, experienced, skilled, espé, tivéc, Clem, Ad. Adv. -uévaec. Eidnoév, Ep. inf. fut. for eidévat, of eidévat, of eidévat. Eidotc, ewe. 7, a thing, Strab.: from Eidorolée, 6v, (eldoc, rotéw) specific, characteristic of a species. Eldoé, coc, 76, (*eldw) that which is seen, the form, shape, figure, Lat. species; freq. of human form in Hom., who usu. has the acc. eldoé dptoroc, ayntoc, Kakdc, GAéyKtog, buoLoe, ete.; sometimes opp. to the understanding, sometimes to bodily strength, v. Od. 17, 454, Il. 21, 316: also of the appearance, look, as of a dog, Od. 17, 308, cf. déuac. Esp. beautiful form, like Lat. forma, Hdt. 1, 199; 8, 105, ete. In Trag. periphr. for the person, Soph. El. 1177.—Il. in genl., a form, figure, fashion, sort, particular kind, cidea, Tov KvBwv, Hdt.'1, 94, eidoc vdcov, Thue. 2, 50, etce.: esp. species, opp. to yévoc, genus, hence also = idéa, Plat., and Arist., cf. Ritter Hist. of Philos. 2, 265, sqq.—III. in later authors ra edn are spices, fine and costly wares. Eldodopéw, O, (eldoc, gépw) to represent, express, Dion. H. EidvAatov, ac, 7, Eidyia, wife of Aescus, Lyc. 1024. EidvAatov, ov, 76, dim. from eldoc strictly a@ little form or image: usu. @ short, highly wrought, descriptive poem, mostly, but by no means only, on pastoral subjects, an idyll, cf. Plin. Ep. 4, 14. EidvAouat,= eiddA2ouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat,= eiddA2ouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. EidvAouat, = eiddAouat, eidaivo pat, Pemp. ap. Stob. p. 46, 9. E A. to see, behold, look at, mostly in aor. eidov, in Hom. and Ep. oft. without augm. Zdov, inf. idezy, in I. and Ep. also idwui, part. idov in Hom. freq. with an adv.,{+~. a,dvra, dypetov lidar, eyeing with astern glance, etc.: he also freq. has more fully 6¢0arpoiorv id. The same act. signf. belongs to the aor. mid. eidéunv, in Hom. more freq. Ep. idwut, inf. idécOaz, subj. idwuat, imperat. idod: with which Hom. has also 6¢6a2,uoi-ovv, or more freq. év 666.,-to see before the eyes: this tense alone is joined with recpdouat, in phrase aye, Te phoouat 768 idwuat, well, I will make trial and see, Od. 6, 126, cf. 21, 159: also without zecpdouar; Just our te look and see, Od. 4, 22; 10, 44. But Hom. also uses both aorists of mental sight, to see, perceive, as must be the case in I. 21,61, dd9a idwyar evi dpe-civ, 708 daeta, ef. ll. 4, 249, Od. 21; 112. This definiteness belongs only to the oldest Greek: in later poets to perceive by any of the senses, Jac. A. P. p. 189. In construction, idea and idéfae are either absol., or used c. acc. followed by a relative clause, where the relative is to be resolved by 67z, so that the “acc. is not strictly the object, but belongs to the verb in the relat. clause, e. g. Od. 10, 195, eldov..vicov, tHv mépt movToc éoteddvura:, i. e. eidov éTt TEept vyoov ovToc éoreduivwtat; ‘though in the remarkable passage, GAbyxov...odTt yap Ide, he saw, i. e. enjoyed not the favor of his spouse, Il. 11, 243, ydéprv is the object; (this phrase must not be confounded with xapv eidéva, v. infr.): freq. also ideiv & Tt, more rare éxé Te, Il. 23, 143, and mpéc Tu, Od. 12, 244, to look at or towards a thing. The imperat. mid. (dod, see, occurring first in Att., is mostly used as an exclamation, lo! behold! Lat. ecce: but it is then written (dod, or sometimes idoy. where it is a true imperat. it remains idod, e.g. idod we, Eur. Hec. 808. Opdw is used as pres., épaxa as perf., dwouat as fut. (for eldfow belongs to signf. B, to know.) But to the signf. to see, belong—II. the Ep. and Ton. pass. and mid. eldouac: aor elodunv, in Hom. also éevodunv, ao, arc, In pass. signf., to be seen, appear, seem, eideTat Hap, doTpa, the day, the stars are visible, appear, Il. 8, 555, cf. 24, 319, Od. 5, 283: metaph., 76 Oé tor Kp elderar elvar, for that seems unto thee to be very death, that is very death in thine eyes, Il.-1, 228, cf. Od..9, 11, etc.; and feq aira "Oyce Képdlov eicato Oud: hence— Z. to have the appearance or look of a thing, take the appearance, make a show of a thing, eica7’ iwev é¢ Ajuvor, he made a show of going to Lemnos, Od. 8, 283; eloato, Oc te furor, it had the look as of a shield, Od. 5, 281; and c. dat., to make one’s self like, be like, eicato foyynv TloAirn, she made herself like Polites in voice, Il. 2, 791, cf 20, 81. Most usu. in part. pres. and aor., eidéuevoc, eioduevoc, fetodjtevoc, besides which Hom. uses only 3 sing. pres. and aor.; and once 2 sing. and 3 plur. aor. An impf. eels he was seen, occurs first in Ap. B. to know: which sign. comes from the perf., for what one has seen or observed, that one knows: hence the word is mostly used of mediate knowledge, whilst for such as is immediate, cavalleria is most usu., Wolf Dem. 461, 2. The tenses which belong to this sign are these: perf. used as pres., oida (in Alcae. 94 e, p. 72, dida) I know, c. part. eidde, inf. eidévae, Ep. tduevac and iduev, imperat. icf. sabj. eidd, Ep. also idéw, opt. eideiny: plapf. as imperf. noe and jdea, Att.-ndy, I knew: fut. eicouat, more rarely and mostly Ep. eidjow (also in Hat. 7, 234): aor. and perf..are supplied from yeyvd- oxw: though in later Greek we have an aor. eiéjaat, Arist. Magn. Mor. 1, 1,3, tc. The forms are so irreg. in pres. ard impf, that they can only be fullyicated of in grammars. In Hom., ica, and Dor., oida¢ is 2 sing. perf. for vic@a, e G. Od. 1, 337, (in Att. also sometimes oticac, Cratin. Malth. 10, cf. Meineke Menand. p. 122: iduev 1 pl. for icwev: besides iSuevar and tduev, inf. for eidévar: idéa subj. for eld, Il. 14, 235. eidere 2 pl. subj. for eld, Od. 9, 17, eidere uev for eldduer’, Il. 1, 363, and idvia fem. part. for eldvia, but only in phrase idviqat zparidecat: plapf. 2 and 3 sing. 7eidyc, jeidn for ydne, Hon. Il. 22, 280, Od. 9, 206, 3 pl. tcav for yaar, IL 18, 405, Od. 4, 772; yaar, Eur. Cyel. 231. n both futures, yet e/djow only in Il. 1, 546, Ep. int. eligoeuev, Od. 6, 237, where it almost passes Into signf. A, to see, and so in the hymns. For the rest v. Buttm. Ausf. Gr. § 199, 111, and Catal. in voc. [icacx has usu. f, as Od. 2, 211, but sometimes also j, in arsis, as Od. 2, 283.] -In Hom. it must be rendered sometimes by to know, understand, have knowledge of, sometimes by to know, discern, perceive; later to come to know, learn; though it may be so taken however in Od. 2, 16: very freq. strenethd. by ed or odga, esp. ev _ olda, I know well, and Part. et eiddc, also eb icOc, know well, be assured. It is often followed by a clause with cic, éxwe or 6rt, and, in case of doubt, with eZ, whether, rarely with the relat. Also followed by acc., or in-fin. Hom. has the peculiar usage, vuata, pndea olde, he is knowing, skilled in wise counsels; and so still more free with ads., memvuuéva, Keyaplouéva, gira, UpTla, iTLa, Ked-ya, GOeuiorra eidévat, but usu. in nart. e{ddac. In this sign, to be skilled im, the word also takes a gen. in Hom., mostly indeed c. part., e. g. rozuv ev eiddc, cunning with the pets ol TEKTOOUVaWY, wayNC, ELC.; but #: in pres. indic., it 15, 412. The inoperat. is freq. in protestations. Lastly Hom. uses, EIEN like totw Zetc, icrw vir Zeve, let Jove know it, be witness, Hom.; Dor. ittw Z.: ydpiv eidévat tevi, to acknowledge a debt to another, thank him, first in Il. 14, 235, Hat. 3, 21, but most freq. in Att., and prose. Post-Hom., usages:—1. to be in a condition, be able, have the power, c. inf., Jac. Anth. 2, 1, p. 308—2. oid’ 671, oic@ dre, used absol. parenthetically as a particle of affirmation, J know, ou know it well, Wolf Dem, 508, 17, eind. Plat. Gorg. 486 B.—3. olc@’ ody; freq. interrog. form, usu. answered by oo’ olda, Valck. Hipp. 598. —4. olc#’ 671, also oicf’ 6 and oic’ @¢, followed by imperat., gives a command without specifying what, as if this was known before, esp. oia@’ 6 dpadcov, for dpdcov, oic@’ 6, v. sub dpdo. (The word always has the digamma in Hom., Fidov, Fede Ete., which remains in Lat. videre, Sanskrit. vid scire, Germ. wissen, our to wit or wot. On the difference of eidwaver from ywyvékerr, V. ywyvéoka, fin. EidwAeiov, ov, 76, (eidwAdrov, bbw) sacrificed to idocs; as subst. 70 eid., Nas EidwAoAarpeia, ac, 7, worship of idols, idolatry, N.T.: and EidwrodarTpéw, G, to worship idols, Eccl.: from EidwAoAdrpnc, ov, 6, 7, (eidwAoAov, | Aarpec) an idol-worshipper, idolater. Eido26u0pdoc, ov, (eidwAov, wopdy) formed after a likeness, like an image, Geop. e EidwAov,ov, 76, (eidoc) ashape, figure, image: in Hom. of disembodied spirits, esp. GpoTdv eidwha KkauovTwy : any unsubstantial form, esp. a vision, phantom, Hom., etc.: hence a phantom of the mind. a fancy, Plat. Phaed. 66 C. —Il. an image in the mind, idea, Xen. Symp. 4, 21: esp. with the Stoics, Cic. Fam. 15, 16.—Il. an image, statue, yuva.xoc, Hat. 1, 51, 6, 58—2. esp. of a god; hence an idol, false god, LXX.—IV. eidwaa odpavia, the constellations, Lat. signa, Ap. Rh. Eidwiorhacréw, 0, to form, model, Heracl. + from EidwiérAactoc, ov, (eidwAoréc) to make an image, eldwAov eid., Plat. Rep. 605 C: to represent by an image or figure, tid, Diod.—2. to body, image forth, depict by words, Longin. Hence y Eidwhoroinate, ewc, 7, « figuring : representation, Sext. imp EidwAoroiéc, ac, 7,=foreg., Plat. Tim. 46 A. EidwAoroiéc, 4, dv, (eidwdo- totoc) of, belonging to figuring or re- presenting, Tévn, Plat.-Soph. 235 A. j sb mad OV; oe ToLéw) figuring, forming, making figures or weer: as subst. 6 e/d., Plat. Soph. 239 D. Eidwrovpyicdc, 9, 6v, (eidwrov, *%nyw)=cldwAoroikoc, Plat. Soph. 266 D. Eldwrovpyicdc, é, (etdwhov, puivo- feat) like an image, Plut.? Eldwrovpyc, é, (eldwrov, yaipw) delighting in idols, Synes. led Elev, Att. 3 plur. opt. from efué, for eijcay, be it so, well, good, proceed, or to proceed, Lat. esto: a very common particle, esp. in Att. dialogue, in passing to the next point, Herm. Eur. Supp. 795: the phrases GAA elev, elév ye, clev Of are more rare. also to express impatience, Ar, Nub. 176. [elev in Att. poets is sometimes used as a spondee, Aesch. Cho. 657, Ar. Pac. 663.] Elevator, opt. aor. 2 act. from type: but eyv, opt. pres. from sip.4 Elevator, adv., (elevate) at once “orth- with, instantly, Al, and Ion. Elevator, inter. J wish! O that! would that! Lat. wtinam! Od. 2, 33: the Dor. ai#e is more free. in Homer: on all OgeAAov and ddeAov, ec, e, ¥. dgeiAw: Cc. opt., of things possible, but not likely; with the past tenses of indic. of things impossible: later also the inf. follows e/@e, Herm. Vig. n. 190, a, cf. sub ed yap. Elevator, E. -iow, poet. for 26iw Elevator, perf. pass. from é6iGw, in the accustomed manner Diog. L. Elevator, Att. for govxa, q. v. Elevator, perf. from the. “ Elevator, ov, 6, (eikac) epith of the Epicureans, because they commemorated their founder’s death. On the twentieth of Gamelion, Ath. 298 D. Eixalo, f.-dow, Att. pass. 7xa-cua, Dind. Ar. Eq. 230, Piers. Moer. p. 182, and on. the eugm. in genl. v. Buttm. Ausf. Gr. § 84, Anm. 3, (e/xéc). To make like to, represent by an image or likeness, portray, Xen. Oec. 10, 1. hence in pass., eleov ypady eixacpe-vy, a figure coloured to the life, Hat. 2, 182; aleroc eixaou., a figure like an eagle, Id. 3, 28; hence—IlI. to liken, compare, Ti Ttvt, Aesch. Cho. 633; ek. TL kai Tl, Hdt. 9, 34, etc.: hence to compare and infer something, to conjecture, guess, Lat. conjicere, esp. im phrase @¢ elxaoat, Hat. 2, 104, et3.: and c. dupl. acc. to guess to be, [Lut. 4, 31, Aesch. Supp. 288, Soph. Ant. Pass. to be like, resemble, revi Eur. Bacch. 942, 1253; also apug tu va, Ar. Ach. 783. Elevacy, inf. of a lengthd. aor. ei Kabov, from etka, to yield, Soph. ete, ; for there is no such pres. es eikdbw, Elmsl. Med. 186, Ellendt Lex. Soph in v. Ei Kai, even though, although, c. indic., Hom. ; c. op é., Il.: distinguished from «ai el by expressing that the thing is really so, Herm, Vig. n. 307: ef. ei C. ElkatoBovalu, ac,7,rashness, Ecel.: from EixaPcvAoe, ov, (elkatoc, Bovan- rash, ill-edwised, Eccl. Eixa:cA5yoc, ov, (eikatoc, Aéyo, talking ut random, Philodem. ap. Vol Herevl. 2, 10. Firacouv0éw, 6, to speak inconsiderately; and Kixaouidia, ac, h, thoughtless talking, ususeless babble; from Elkar6uddoc, ov, (sixaioc, wiboc, talking at random or to no purpose, Eccl. 5 Hikacoppnuoovvn, G,=elkavopvbew, Eixaroppnuoovvn, n¢ = elkacouv- Gia: from é Elxacoppuorv, ov, gen. ov, (elkat oc, pyua)=elixacouvboc. Bixaioc, ata, aiov, without plan purpose: random, rash, hasty, nearly = Lat. temerarius, Soph. Fr. 288.— IL—rvyeév, casual, hence common, worthless, Luc. Adv. -wc, Joseph Hence. Eixavoctvn ne, 7, thoughtlessness Timon ap. Diog. L, 5 11. 399 EIKO Bixarétys, ntoc, 7, =fcreg., Philo- | warring against images, assaulting im- dcm. ap. Vol. Hercul. 2. 9. Eikdc, Gdoc, 7, (elkoor) the number twenty, for elxotde.—Il. the twentieth day of the month, sub. 7uépa, Hes. Op. 790, 818: also pl. eladdec, Ar. Nub. 17. One of the days of the Eleusinian mysteries was also so called, Eur. Ion 1076. Eixdoa 1 aor. inf. act. from ela- Q. “Hixdodw, Aeol. and Dor. for elxa- $a, Sapph. 34. Wixdoxia, ac, 7, (elkafw) a likeness, image, representation, Xen.—II. a comparison, Plut.: a conjecture, a guess-ing, Plat. Rep. 534 A.: Rixacua, atoc, 76, (elxdlw) a likeness, image, Aesch. Theb. 523. Eixaopoc, od, 6, (elkalw) one who conjectures, a guesser, diviner, TOV weA-A6vrwy, Thue. 1, 138. Hixaorexde, 4, 6v, (eixdla) of, belonging, suited to representing, guessing, OF interpreting: 7 eik., Sub. Téyvn, the art of copying or portraying, Plat. Soph. 235 D, ete.: rd eix., sub. ér(bpyuura, aduerbs of doubting. Adv. ‘KOC, by conjecture, by guessing, Elkaarée, %, bv, (e1xagw) to be compared, like, Soph. T. 699: copied, represented, i; Eixdre, Dor. for elxoae. Elke,