对于cf和swing算法的介绍可以参考ItemCF的演进：狭义 VS 广义

基于cf的推荐相似度

这里介绍这样一个场景，我们有了大量的电商购买数据，希望通过cf算法计算不同的类目之间的相似度，以方便对用户购买进行兴趣探索。
使用SQL实现需要通过join来完成user和cate构成矩阵计算cate之间的余弦相似度，公式如下，而这里又分成三步
$\cos (A,B)=\frac{a_1{b}_1+a_2{b}_2+\dots +a_n{b}_n}{\sqrt{(a_1{a}_1+a_2{a}_2+\dots +a_n{a}_n)(b_1{b}_1+b_2{b}_2+\dots +b_n\ast b_n)}}$

1. user_cate获得user对每个cate的score：普通的score可以设计为用户的订单量，但这会导致高活跃用户对相似度计算贡献太多导致失真，因此这里设计score为用户订单量对数的倒数，即对高活跃用户对相似度的贡献进行惩罚；同时为了防止高热类目计算量过大，使用随机采样保证每个类目下订单量少于10000000
2. cc_prod计算同user下的cate交叉score乘积，里面包含余弦相似度矩阵的分子（即内积）和分母（即模）
3. 最后计算余弦相似度，内积除以两个模

with user_cate as (
    select
        t1.user_id,
        t1.cate_id,
        sum(1 / log(t2.act + 1)) as score
    from (
        select
            user_id,
            category_id as cate_id,
            row_number() over(partition by category_id order by rand(47)) as rn
        from
            project.my_table
        where
            p_date > '{{ds_nodash - 30}}' and p_date <= '{{ds_nodash}}'
            and user_id > 0
            and category_id > 0
            and pay_order_cnt > 0
    ) t1
    join (
        select
            user_id,
            count(1) as act
        from
            project.my_table
        where
            p_date > '{{ds_nodash - 30}}' and p_date <= '{{ds_nodash}}'
            and user_id > 0
            and category_id > 0
            and pay_order_cnt > 0
        group by user_id
    ) t2 on t1.user_id = t2.user_id
    where t1.rn < 10000000
    group by t1.user_id,
        t1.cate_id
),
cc_prod as (
    SELECT cate1, cate2, sum(score_prod) AS prod
    FROM (
        SELECT t1.cate_id AS cate1, t2.cate_id AS cate2, t1.score*t2.score as score_prod
        FROM user_cate t1
        JOIN user_cate t2
        ON t1.user_id = t2.user_id
    ) t
    GROUP BY cate1, cate2
)

SELECT t1.cate1, t1.cate2, t1.prod/SQRT(t2.prod*t3.prod) AS sim 
FROM cc_prod t1
JOIN cc_prod t2 ON t1.cate1=t2.cate1 AND t1.cate1=t2.cate2
JOIN cc_prod t3 ON t1.cate2=t3.cate1 AND t1.cate2=t3.cate2
where t1.cate1 != t1.cate2

基于swing的推荐物品相似度

swing算法的核心公式如下，其中$U_i$表示购买类目$i$的用户集合，$U_j$表示购买类目$j$的用户集合，$I_u$表示用户$u$购买的类目集合，$I_v$表示用户$v$购买的类目集合，具体算法解释可以参考swing：一种工业界广泛使用的召回算法，用SQL实现时也分为三步

1. user_cate获得user和cate建立的边关系，注意这里不考虑订单量了；这里测试发现cate能接受的数据量最多采样为20000，再多就基本跑不动了
2. uu_cate计算任意两个user共同购买cate的配对
3. 最后计算任意两个user共同购买的任意两个cate的配对（t1）和任意两个user共同购买cate的数量cate_cnt（t2），将1 / (1 + cate_cnt)在任意两个cate的配对里累加，即为最后的cate相似度

with user_cate as (
    select
        user_id,
        cate_id
    from (
        select
            user_id,
            category_id as cate_id,
            row_number() over(partition by category_id order by rand(47)) as rn
        from
            project.my_table
        where
            p_date = '{{ds_nodash}}'
            and user_id > 0
            and category_id > 0
            and pay_order_cnt > 0
    ) t
    where rn < 20000
    group by user_id,
        cate_id
),
uu_cate as (
	select
	    uc1.user_id as u1,
	    uc2.user_id as u2,
	    uc1.cate_id as cate
	from user_cate as uc1
	left join user_cate as uc2
	on uc1.cate_id = uc2.cate_id
	where uc1.user_id != uc2.user_id
)

select 
    cate1,
    cate2,
    sum(1 / (1 + cate_cnt)) as sim
from (
    select 
        uuc1.u1,
        uuc1.u2,
        uuc1.cate as cate1,
        uuc2.cate as cate2
    from uu_cate as uuc1
    left join uu_cate as uuc2
    on uuc1.u1 = uuc2.u1 and uuc1.u2 = uuc2.u2
    where uuc1.cate != uuc2.cate
) t1
left join (
    select
        u1,
        u2,
        count(cate) as cate_cnt
    from uu_cate
    group by u1, u2
) t2 on t1.u1 = t2.u1 and t1.u2 = t2.u2
group by cate1, cate2