推荐系统：理论、方法与评估

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"这篇综述文章全面探讨了推荐系统，涵盖了包括SVD（奇异值分解）、ITEM-CF（基于物品的协同过滤）和USER-CF（基于用户的协同过滤）在内的多种推荐算法，以及用于评估推荐系统性能的各种指标。作者来自不同学术背景，包括中国杭州师范大学的信息经济研究所、瑞士弗里堡大学的物理系、中国电子科技大学的网络科学中心、英国阿斯顿大学的非线性和复杂性研究小组以及北京计算科学研究中心。文章旨在弥补现有研究中对各种推荐方法统一和比较不足的问题，回顾并分析了该领域的最新进展。" 推荐系统是一种用于处理海量信息过滤的技术，随着互联网的快速发展，其重要性日益凸显。该文主要关注以下几个方面： 1. **协同过滤**：协同过滤是推荐系统中最常见的技术之一，分为基于用户的协同过滤（USER-CF）和基于物品的协同过滤（ITEM-CF）。USER-CF通过分析用户的历史行为，找出具有相似兴趣的用户，然后将一个用户未曾体验但其他相似用户喜欢的项目推荐给他。而ITEM-CF则是通过分析用户对物品的评价，找出物品之间的相似性，然后向用户推荐他们可能喜欢的相似物品。 2. **奇异值分解（SVD）**：在推荐系统中，SVD是一种矩阵分解技术，用于挖掘用户-物品评分矩阵中的潜在结构。通过将矩阵分解为低秩近似，SVD可以捕捉用户和物品之间的隐藏关系，并据此进行预测，从而推荐未被用户评价的物品。 3. **评价指标**：推荐系统的性能通常通过一系列评价指标来衡量，如准确率、召回率、F1分数、覆盖率、多样性等。准确率和召回率关注推荐列表中正确项目的比例，而覆盖率衡量推荐系统能否覆盖整个物品库。多样性则强调推荐结果的不重复性，以提供更丰富的用户体验。 4. **多学科交叉研究**：推荐系统的研究不仅仅是计算机科学的领域，还吸引了社会科学家、物理学家等多学科研究人员的参与。这表明推荐系统的研究正逐渐成为跨学科的热点，有助于推动理论与实践的结合，促进新的发现和方法。 5. **领域进展**：文章回顾了推荐系统领域的最新发展，这可能包括深度学习的应用、混合推荐方法、社交网络影响的研究以及个性化推荐的挑战等。这些进展旨在提高推荐的准确性和用户满意度，同时解决冷启动和稀疏数据等问题。这篇综述文章为读者提供了推荐系统领域的全面视图，不仅讨论了核心算法，还涵盖了评估标准和研究趋势，对于深入理解和进一步研究推荐系统具有很高的参考价值。

Age

Sex

Nationality

Job

...

Profiles

Users Items

Attributes

Contents

Category

Author

Year

...

Harry is a

very special

child

...

I like it

It is OK

I dislike it

I like it

Figure 4: Illustration of a recommender system consisted of ﬁve users and four books. The basic infor-

mation contained by every recommender system is the relations between users and objects that can be

represented by a bipartite graph. This illustration also exhibits some additional information frequently

exploited in the design of recommendation algorithms, including user proﬁles, object attributes and object

content.

data. Let M be the number of users and let N be the number of all objects that can

be evaluated and recommended. Note that object is simply as a generic term which can

represent books, movies, or any other kind of consumed content. To stay in line with

standard terminology, we sometimes use item which has the same meaning. To make the

notation more clear, we restrict to Latin indices i and j when enumerating the users and

to Greek indices α and β when enumerating the objects. Evaluation/rating of object α

by user i is denoted as r

iα

. This evaluation is often numerical in an integer rating scale

(think of Amazon’s ﬁve stars)—in this case we speak of explicit ratings. Note that the

common case of binary ratings (like/dislike or good/bad) also belongs to this category.

When objects are only collected (as in bookmark sharing systems) or simply consumed

(as in online newspaper or magazine without rating systems) or when “like” is the only

possible expression (as on Facebook), we are left with unary ratings. In this case, r

iα

= 1

represents a collected/consumed/liked object and r

iα

= 0 represents a non-existing evalua-

tion (See Fig. 4). Inferring users’ conﬁdence levels of ratings is not a trival task, especially

from the binary or unary ratings. Accessorial information about users’ behavior may be

helpful, for example, the users’ conﬁdence levels can be estimated by their watching time

of television shows and with the help of this information, the quality of recommendation

can be improved [93]. Even if we have explict ratings, it does not mean we know how and

why people vote with these ratings–Do they have standards of numerical ratings or they

just use ratings to present orders? Recent evidence [94] to some extent supports the latter

ansatz.

Alice Bob Carol

Titanic 5 1 5

2001: A Space Odyssey 1 5 2

Casablanca 4 2 ?

Table 2: Recommendation process in a nutshell: to estimate the potential favorable opinion of Carol about

Casablanca, one can use the similarity of her with those of Alice. Alternatively, one can note that ratings

of Titanic and Casablanca follow a similar pattern, suggesting that people who liked the former might also

like the latter.

The goal of a recommender system is to deliver lists of personalized “recommended”

objects to its users. To this end, evaluations can be predicted or, alternatively, recommen-

dation scores can be assigned to objects yet unknown to a given user. Objects with the

highest predicted ratings or the highest recommendation scores then constitue the recom-

mendation list that is presented to the target user. There is an extensive set of performance

metrics that can be used to evaluate the resulting recommendation lists (see Sec. 3.4). The

usual classiﬁcations of recommender systems is as follows [15]:

1. Content-based recommendations: Recommended objects are those with content sim-

ilar to the content of previously preferred objects of a target user. We present them

in Sec. 4.2.3.

2. Collaborative recommendations: Recommended objects are selected on the basis of

past evaluations of a large group of users. They can be divided into:

(a) Memory-based collaborative ﬁltering: Recommended objects are those that were

preferred by users who share similar preferences as the target user, or, those that

are similar to the other objects preferred by the target user. We present them

in Sec. 4 (Standard similarity-based methods) and Sec. 7 (methods employing

social ﬁltering).

(b) Model-based collaborative ﬁltering: Recommended objects are selected on mod-

els that are trained to identify patterns in the input data. We present them in

Sections 5 (dimensionality reduction methods) and 6 (diﬀusion-based methods).

3. Hybrid approaches: These methods combine collaborative with content-based meth-

ods or with diﬀerent variants of other collaborative methods. We present them in

Sec. 8.4.

3.4. Evaluation Metrics for Recommendation

Given a target user i, a recommender system will sort all i’s uncollected objects and

recommend the top-ranked objects. To evaluate recommendation algorithms, the data is

usually divided into two parts: The training set E

and the probe set E

. The training

set is treated as known information, while no information from the probe set is allowed

to be used for recommendation. In this section we brieﬂy review basic metrics that are

used to measure the quality of recommendations. How to choose a particular metric (or

metrics) to evaluate recommendation performance depends on the goals that the system is

supposed to fulﬁll. Of course, the ultimate evaluation of any recommender system is given

by the judgement of its users.

3.4.1. Accuracy Metrics

Rating Accuracy Metrics. The main purpose of recommender systems is to predict users’

future likes and interests. A multitude of metrics exist to measure various aspects of

recommendation performance. Two notable metrics, Mean Absolute Error (MAE) and

Root Mean Squared Error (RMSE), are used to measure the closeness of predicted ratings

to the true ratings. If r

uα

is the true rating on object α by user i, ˜r

iα

is the predicted

rating and E

is the set of hidden user-object ratings, MAE and RMSE are deﬁned as

MAE =

(i,α)∈E

iα

− ˜r

iα

|, (8)

RMSE =



(i,α)∈E

iα

− ˜r

iα

)



1/2

. (9)

Lower MAE and RMSE correspond to higher prediction accuracy. Since RMSE squares the

error before summing it, it tends to penalize large errors more heavily. As these metrics

treat all ratings equally no matter what their positions are in the recommendation list,

they are not optimal for some common tasks such as ﬁnding a small number of objects

that are likely to be appreciated by a given user (Finding Good Objects). Yet, due to their

simplicity, RMSE and MAE are widely used in the evaluation of recommender systems.

Rating and Ranking Correlations. Another way to evaluate the prediction accuracy is to

calculate the correlation between the predicted and the true ratings. There are three well-

known correlation measures, namely the Pearson product-moment correlation [95], the

Spearman [96] correlation and Kendall’s Tau [97]. The Pearson correlation measures the

extent to which a linear relationship is present between the two sets of ratings. It is deﬁned

P CC =

(˜r

−

˜r)(r

− ¯r)

(˜r

−

˜r)

− ¯r)

, (10)

where r

and ˜r

are the true and predicted ratings, respectively. The Spearman correlation

coeﬀcient ρ is deﬁned in the same manner as the Pearson correlation, except that r

and ˜r

are replaced by the ranks of the respective objects. Similarly to the Spearman correlation,

Kendall’s Tau also measures the extent to which the two rankings agree on the exact values

of ratings. It is deﬁned as τ = (C − D)/(C + D) where C is the number of concordant

pairs—pairs of objects that the system predicts in the correct ranked order and D is the

number of discordant pairs—pairs that the system predicts in the wrong order. τ = 1

when the true and predicted ranking are identical and τ = −1 when they are exactly

opposite. For the case when objects with equal true or predicted ratings exist, a variation

of Kendall’s Tau was proposed in [13]

τ ≈

C − D

(C + D + S

)(C + D + S

)

, (11)

where S

is the number of object pairs for which the true ratings are the same, and S

is the number of object pairs for which the predicted ratings are the same. Kendall’s Tau

metric applies equal weight to any interchange of successively ranked objects, no matter

where it occurs. However, interchanges at diﬀerent places, for example between top 1 and

2, and between 100 and 101, may have diﬀerent impacts. Thus a possible improved metric

could give more weight to object pairs at the top of the true ranking.

Similar to Kendall’s Tau, the normalized distance-based performance measure (NDPM)

was originally proposed by Yao [98] to compare two diﬀerent weakly ordered rankings. It

is based on counting the number of contradictory pairs C

−

(for which the two rankings

disagree) and compatible pairs C

(for which one ranking reports a tie while the other

reports strict preference of one object over the other). Denoting the total number of strict

preference relationships in the true ranking as C, NDPM is deﬁned as

NDP M =

−

+ C

. (12)

Since this metric does not punish the situation where the true ranks are tied, it is more

appropriate than correlation metrics for domains where users are interested in objects that

are good-enough.

Classiﬁcation Accuracy Metrics. Classiﬁcation metrics are appropriate for tasks such as

“Finding Good Objects”, especially when only implicit ratings are available (i.e., we know

which objects were favored by a user but not how much they were favored). When a

ranked list of objects is given, the threshold for recommendations is ambiguous or variable.

To evaluate this kind of systems, one popular metric is AUC (Area Under ROC Curve),

where ROC stands for the receiver operating characteristic [99] (for how to draw a ROC

curve see [13]). AUC attempts to measure how a recommender system can successfully

distinguish the relevant objects (those appreciated by a user) from the irrelevant objects (all

the others). The simplest way to calculate AUC is by comparing the probability that the

relevant objects will be recommended with that of the irrelevant objects. For n independent

comparisons (each comparison refers to choosing one relevant and one irrelevant object),

if there are n

times when the relevant object has higher score than the irrelevant and n

times when the scores are equal, then according to [100]

AUC =

+ 0.5n

. (13)

Clearly, if all relevant objects have higher score than irrelevant objects, AUC = 1 which

means a perfect recommendation list. For a randomly ranked recommendation list, AUC =

0.5. Therefore, the degree of which AUC exceeds 0.5 indicates the ability of a recommen-

dation algorithm to identify relevant objects. Similar to AUC is a so-called Ranking Score

proposed in [101]. For a given user, we measure the relative ranking of a relevant object in

this user’s recommendation list: when there are o objects to be recommended, a relevant

object with ranking r has the relative ranking r/o. By averaging over all users and their

relevant objects, we obtain the mean ranking score RS—the smaller the ranking score, the

higher the algorithm’s accuracy, and vice versa.

Since real users are usually concerned only with the top part of the recommendation

list, a more practical approach is to consider the number of a user’s relevant objects ranked

in the top-L places. Precision and recall are the most popular metrics based on this. For

a target user i, precision and recall of recommendation, P

(L) and R

(L), are deﬁned as

(L) =

(L)

, R

(L) =

(L)

(14)

where d

(L) indicates the number of relevant objects (objects collected by i that are present

in the probe set) in the top-L places of the recommendation list, and D

is the total number

of i’s relevant objects. Averaging the individual precision and recall over all users with

at least one relevant object, we obtain the mean precision and recall, P (L) and R(L),

respectively. These values can be compared with precision and recall resulting from random

recommendation, leading to precision and recall enhancements as deﬁned in [37]

(L) = P (L)

, e

(L) = R(L)

, (15)

where M and N are the number of users and objects, respectively, and D is the total

number of relevant objects. While precision usually decreases with L, recall always grows

with L. One may combine them into a less L-dependent metric [102, 103]

(L) =

2P R

P + R

(16)

which is called F

-score. Many other measurements which combine precision and recall are

used to evaluate the eﬀectiveness of information retrieval, but rarely applied to evaluate

recommendation algorithms: Average Precision, Precision-at-Depth, R-Precision, Recipro-

cal Rank [104], Binary Preference Measure [105]. A detailed introduction and discussion

of each combination index can be found in [106].

3.4.2. Rank-weighted Indexes

Since users have limited patience on inspecting individual objects in the recommended

lists, user satisfaction is best measured by taking into account the position of each relevant

object and assign weights to them accordingly. Here we introduce three representative in-

dexes that follow this approach. For a detailed discussion of their strengths and weaknesses

see [13].

Half-life Utility. The half-life utility metric attempts to evaluate the utility of a recommen-

dation list to a user. It is based on the assumption that the likelihood that a user examines

a recommended object decays exponentially with the object’s ranking. The expected utility

of recommendations given to user i hence becomes [107]

α=1

max(r

iα

− d, 0)

iα

−1)/(h−1)

, (17)

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