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+arXiv:2301.12159v1  [cs.CV]  28 Jan 2023
+ClusterFuG: Clustering Fully connected Graphs by Multicut
+Ahmed Abbas 1 Paul Swoboda 1 2
+Abstract
+We propose a graph clustering formulation based
+on multicut (a.k.a. weighted correlation cluster-
+ing) on the complete graph.
+Our formulation
+does not need specification of the graph topology
+as in the original sparse formulation of multicut,
+making our approach simpler and potentially bet-
+ter performing. In contrast to unweighted corre-
+lation clustering we allow for a more expressive
+weighted cost structure. In dense multicut, the
+clustering objective is given in a factorized form
+as inner products of node feature vectors. This al-
+lows for an efficient formulation and inference in
+contrast to multicut/weighted correlation cluster-
+ing, which has at least quadratic representation
+and computation complexity when working on
+the complete graph. We show how to rewrite clas-
+sical greedy algorithms for multicut in our dense
+setting and how to modify them for greater ef-
+ficiency and solution quality. In particular, our
+algorithms scale to graphs with tens of thousands
+of nodes. Empirical evidence on instance seg-
+mentation on Cityscapes and clustering of Ima-
+geNet datasets shows the merits of our approach.
+1. Introduction
+Graph-based clustering approaches, primarily among them
+multicut (Chopra & Rao, 1993), are theoretically appeal-
+ing: They do not need specification of the number of clus-
+ters, but infer them as part of the optimization process.
+They allow for a flexible clustering objective with attrac-
+tive and repulsive costs between pairs of nodes.
+They
+are also theoretically well-understood as optimization prob-
+lems with intensively studied polyhedral descriptions. Effi-
+cient solvers that scale well and give high quality solutions
+have also been developed.
+As a drawback, graph-based clustering approaches need
+specification of the underlying graph topology. In prac-
+1MPI for Informatics, Saarland Informatics Campus, Germany
+2University of Mannheim, Germany. Correspondence to: Ahmed
+Abbas <ahmed.abbas@mpi-inf.mpg.de>.
+Preprint.
+tice, this means an additional engineering effort as well as
+the possibility to not get it right, which would decrease the
+downstream task performance. Naively circumventing this
+challenge by using the complete graph is not scalable – the
+number of edges grows quadratically. One approach to re-
+solve this conundrum is graph structure learning, which in-
+fers the graph topology as part of the inference process, but
+adds considerable additional complexity.
+We propose a method to solve graph clustering efficiently
+on complete graphs. Our formulation will use the well-
+known edge-based multicut formulation and only restrict
+the way edge costs can be computed: they need to be based
+on inner products of node features. This has two advan-
+tages: First, it reduces storage requirements. Instead of
+storing a full adjacency matrix of edge costs as in multicut,
+which grows quadratically with the number of nodes, we
+only need to store a linear number of node features and can
+compute edge costs on demand. Second, operations needed
+in multicut algorithms can be made scalable. Instead of
+operating on the complete graph we can sparsify it adap-
+tively during the solving process. This allows to simulate
+the workings of multicut algorithms on complete graphs by
+working on a small subset of it. The key technical ingre-
+dient to obtain these sparse subgraphs will be fast nearest
+neighbor search, for which efficient and scalable implemen-
+tations exist (Johnson et al., 2019). In effect, this allows us
+to solve large dense multicut instances in moderate time,
+which is not possible with existing solvers. In detail, our
+contribution is as follows:
+Formulation: We propose multicut on complete graphs
+with factorized edge costs as an efficiently repre-
+sentable graph clustering formalism.
+Algorithm: We propose scalable algorithms for solving
+the dense multicut problems, one mimicking exactly
+the original greedy additive edge constraction (GAEC)
+algorithm (Keuper et al., 2015), the other a more effi-
+cient variant in the spirit of the balanced edge contrac-
+tion heuristic (Kardoost & Keuper, 2018)1.
+Empirical: We show efficacy in terms of memory and run-
+time of our solvers and show the merit of using them
+1Our
+code
+is
+available
+at
+https://github.com/aabbas90/cluster-fug
+
+Clustering Fully connected Graphs by Multicut
+for image segmentation on Cityscapes and clustering
+of ImageNet classification dataset.
+2. Related work
+Multicut and correlation clustering:
+The original mul-
+ticut problem is formulated as an extension of the min-
+cut problem to multiple terminals with non-negative edge
+costs (Hu, 1963).
+In machine learning the multicut
+problem is defined differently and is equivalent (up to
+variable involution) to the correlation clustering prob-
+lem (Demaine et al., 2006), i.e. arbitrary edges costs and
+no terminals. For the purpose of this work we will use the
+latter definition of multicut. The polyhedral geometry of
+the multicut problem has been studied in (Deza et al., 1992;
+Chopra & Rao, 1993; Oosten et al., 2001).
+Although the multicut problem is NP-Hard (Bansal et al.,
+2004; Demaine et al., 2006), greedy algorithms perform
+well in practice for computer vision and machine learn-
+ing tasks (Keuper et al., 2015; Levinkov et al., 2017;
+Bailoni et al., 2022).
+More involved algorithms in-
+clude message passing in the dual domain for multi-
+cut, studied in (Swoboda & Andres, 2017; Lange et al.,
+2018; Abbas & Swoboda, 2022).
+These algorithms give
+lower bounds and improved primal solutions.
+Another
+line of efficient primal heuristics is based on move-
+making (Beier et al., 2014; 2015). All these graphs, while
+efficient, scale with the number of edges, making them
+unsuitable for very large dense graphs.
+Algorithms for
+correlation clustering on complete graphs were proposed
+in (Pan et al., 2015; Veldt, 2022). However, they only al-
+low unweighted edges. In this paper we consider efficient
+algorithms on full graphs and with weighted edges.
+K-Means:
+The K-means problem (Lloyd, 1982) is sim-
+ilar to our approach in that it works directly on feature
+representations and its objective is based on L2-distances
+between features. Similarly to our algorithm, large num-
+ber of points are handled by efficiently computing kNN-
+graphs (Qaddoura et al., 2020), thereby reducing run time.
+In contrast to multicut, the number of clusters must be
+given a-priori, while in multicut it is derived as part of the
+optimization process.
+Other clustering approaches:
+There are a number of
+other paradigms for clustering. A prominent approach is
+spectral clustering, in which a weighted graph is given and
+a clustering is computed with the help of the eigenvec-
+tors of the graph Laplacian (Von Luxburg, 2007; Jia et al.,
+2014). The work (Dhillon et al., 2007) shows connections
+between weighted k-means and multiple spectral clustering
+approaches. As for K-means and unlike multicut, spectral
+clustering requires the number of clusters to be specified.
+i
+j
+fi fj
+(0, 0)
+Figure 1: Example illustration of dense multicut prob-
+lem (3) on 5 nodes. Each node i is associated with a vec-
+tor fi ∈ R2 and all possible edges between distinct nodes
+are considered (i.e. the complete graph). The edge cost be-
+tween a pair of nodes i, j is measured by ⟨fi, fj⟩ and attrac-
+tive/repulsive edges are colored green/red. Edge thickness
+represents absolute edge cost. Also shown is the optimal
+partitioning to 2 clusters with cut edges denoted by dashed
+lines.
+3. Method
+A decomposition (or clustering) of a weighted graph G =
+(V, E, c) with vertices V , edges E and edge costs c ∈ RE
+can be obtained by solving the following multicut problem
+min
+y∈MG
+�
+ij∈E
+cijyij.
+(1)
+We say that an edge ij with cij > 0 is attractive. Its end-
+points prefer to be in the same cluster. In the opposite case
+cij < 0 we call the edge repulsive. The set MG enumer-
+ates all possible partitions of G defined as
+MG =
+
+
+δ(V1, . . . , Vn) :
+n ∈ N
+Vi ∩ Vj = ∅
+∀i ̸= j
+V1 ˙∪ . . . ˙∪Vn = V
+
+
+ .
+(2)
+where δ(·, . . . , ·) ⊆ E is the set of edges straddling distinct
+components.
+The goal of our work is to consider the scenario when the
+graph G is complete i.e. E = {ij : i ∈ V, j ∈ V \ {i}}.
+For large graphs storage and processing of edge costs c be-
+comes prohibitive. To address this issue we instead require
+as input a feature vector fi ∈ Rd for each node i in V . The
+edge costs between a pair of nodes i and j can then be mea-
+sured on-demand through some function s(fi, fj) → R. In
+this case the multicut problem becomes
+min
+y∈MG
+�
+i∈V
+�
+j∈V \i
+s(fi, fj)yij,
+(3)
+
+Clustering Fully connected Graphs by Multicut
+which we term as dense multicut problem. An illustration
+of our formulation is given in Figure 1. In the following
+we first revisit an algorithm to approximately solve (1) and
+show its extensions for dense multicut problem (3).
+3.1. Greedy Additive Edge Contraction
+The
+greedy
+additive
+edge
+contraction
+(GAEC)
+scheme
+(Keuper et al.,
+2015)
+computes
+approximate
+solution of the multicut problem (1) as given in Algo-
+rithm 1. It initializes each node as a separate cluster and
+iteratively contracts a pair of nodes i, j with the largest
+non-negative cost cij (if it exists). Let m be the node i and
+j are contracted to. The edge costs of edges incident to m
+are
+cml = cil + cjl, l ∈ Ni ∪ Nj \ {i, j},
+(4)
+where costs of non-existing edges are assumed to be 0
+and Ni corresponds to neighbours of i in graph G. For
+complete graphs directly applying this algorithm by oper-
+ating on edge costs is computationally expensive. More-
+over, since each node is connected to all other nodes (Ni =
+V \ {i}), cost updates (4) during edge contraction take
+O(|V |) instructions.
+Algorithm 1: GAEC (Keuper et al., 2015)
+Data: Weighted graph G = (V, E, c)
+Result: Clusters V
+1 while maxuv∈E cuv ≥ 0 do
+2
+m := ij = arg maxuv∈E cuv
+// Aggregate edge costs
+3
+cml = cil + cjl, l ∈ Ni ∪ Nj \ {i, j}
+// Update edges
+4
+E = (E∪{ml|l ∈ Ni∪Nj})\{il}l∈Ni ∪{jl}l∈Nj
+// Update nodes
+5
+V = (V ∪ m) \ {i, j}
+Contraction on complete graphs:
+We show how to per-
+form a more efficient (and equivalent) contraction by oper-
+ating on the node features f by our formulation (3) for the
+particular case of s(·, ·) defined as
+s(fi, fj) = ⟨fi, fj⟩.
+(5)
+From now on, unless stated otherwise, our edge costs will
+be given by (5).
+Lemma 3.1 (Contraction with node features). Assume
+edge costs are measured by (5) and nodes i and j are con-
+tracted to m. Then features of node m given by
+fm = fi + fj
+(6)
+produce contracted edge costs according to (4).
+Proof. By applying (5) for l ∈ V and comparing with (4)
+we get
+s(fm, fl) = ⟨fm, fl⟩ = ⟨fi, fl⟩ + ⟨fj, fl⟩
+= s(fi, fl) + s(fj, fl) .
+Next we will build on the previous result to devise heuris-
+tics for solving dense multicut problem (3) efficiently.
+GAEC for complete graphs:
+We devise an algorithm
+which exactly imitates GAEC (Keuper et al., 2015) but
+is applicable to our formulation on complete graphs (3).
+Specifically to make GAEC efficient with node features
+and a complete graph, we sparsify the original graph G by
+working on its directed k-nearest neighbours (NN) graph
+(V, A). The NN graph stores candidate edges for contrac-
+tion.
+The arc set A is populated by nearest neighbour
+search w.r.t. feature similarity (5) and is updated on each
+edge contraction. We denote by N +
+i
+the set of outgoing
+neighbours of i in the NN graph i.e. {l|(l, i) ∈ A} and simi-
+larly by N −
+i the incoming neighbours. Moreover we define
+N +
+ij as N +
+i ∪ N +
+j . The complete strategy to obtain a fea-
+sible solution of dense multicut problem is described in Al-
+gorithm 2. It imitates Algorithm 1 by iteratively searching
+and contracting the most attractive edge, but it restricts its
+search only to the NN graph thereby reducing computation.
+After contraction, the NN graph is updated (lines 5-8) by
+only recomputing nearest neighbors of nodes which were
+affected by the contraction in the NN graph.
+Algorithm 2: Dense GAEC
+Data: Node features fi, ∀i ∈ V ; Number of nearest
+neighbours k
+Result: Clusters V
+// Find nearest neighbours of each node
+1 A = {(i, j)|i ∈ V, j ∈ arg top-ki′̸=i⟨fi, fi′⟩}
+2 while max(u,v)∈A⟨fu, fv⟩ ≥ 0 do
+3
+m := (i, j) = arg top-k(u,v)∈A⟨fu, fv⟩
+// Aggregate node features
+4
+fm = fi + fj
+// Update nodes
+5
+V = (V ∪ m) \ {i, j}
+// Nodes with i, j as NN
+6
+H = {(q, arg maxl∈V \q⟨fm, fl⟩)|q ∈ N −
+ij }
+// NN of merged node
+7
+H = H ∪ {(m, r)|r = arg top-kl∈V \m⟨fm, fl⟩}
+// Update arcs
+8
+A = (A ∪ H) \ {(q, i)}q∈N −
+i ∪ {(q, j)}q∈N −
+j
+Proposition 3.2 (Dense Greedy Contraction). Algorithm 2
+always merges a pair of nodes i and j with the largest edge
+
+Clustering Fully connected Graphs by Multicut
+cost i.e.
+(i, j) ∈ arg max
+(u,v)∈A
+⟨fu, fv⟩ =⇒ ⟨fi, fj⟩ ≥ max
+u,v̸=u⟨fu, fv⟩.
+(7)
+Proof. The statement is trivially satisfied before any merge
+operation is performed since A is constructed by nearest
+neighbour search over all nodes in line 1 of the algorithm.
+We now show that after each merge operation (i.e. after
+line 8 of the algorithm) the statement (7) still holds. We
+define Q = m ∪ {q|q ∈ N −
+ij } to be the set of nodes using i
+or j as their nearest neighbours. Two cases can arise:
+Case 1: {i, j}∩Q ̸= ∅:
+Due to nearest neighbour search
+for all nodes in Q at lines 6 and 7, the statement holds.
+Case 2: {i, j} ∩ Q = ∅:
+In this case if i is the con-
+tracted node m from the last edge contraction operation
+then (i, j) ∈ A due to line 6. If i ̸= m then it remains
+connected to its nearest neighbours either due to the initial
+NN search at line 1 or the NN update at lines 6 and 7.
+The above result guarantees that the most attractive edge
+will always be present in the nearest neighbour graph
+thus foregoing the need to search in the complete graph.
+This proves that the Algorithm 2 performs locally optimal
+merges as proposed in (Keuper et al., 2015) and is also scal-
+able to large complete graphs. As a downside the algorithm
+requires costly nearest neighbour search after every edge
+contraction. Since computing nearest neighbours and con-
+tracting edges is not commutative, in the worst case one
+has to recompute the nearest neighbours on the contracted
+graph from scratch.
+Incremental nearest neighbours:
+For faster nearest
+neighbour updates after edge contraction we show how
+to reuse more of the previously computed nearest neigh-
+bors through the following two approaches. First, for all
+nodes whose nearest neighbours are merging nodes (i.e.
+line 6 of Alg. 2), we check if merged node m is already
+a nearest neighbour without requiring exhaustive search.
+Specifically assume a contracting node i was a k-nearest
+neighbour of some other node q ∈ V \ i.
+Then the
+merged node m is a k-nearest neighbour of q if ⟨fq, fm⟩ ≥
+minl∈N +
+q ⟨fq, fl⟩. This check can be cheaply performed for
+all such nodes thereby reducing computation. Second, we
+devise a criterion which can allow to efficiently populate
+nearest neighbours of the contracted node m.
+Proposition 3.3 (Incremental nearest neighbours). Let the
+k-nearest neighbours N +
+i , N +
+j
+of nodes i and j be given.
+Assume that nodes i, j are merged to form a new node m.
+Then edge costs between nodes v ∈ V \ N +
+ij and m are
+i
+j
+N +
+i
+N +
+j
+N −
+ij
+Figure 2: Illustration of nearest neighbour graph and an
+edge ij being contracted. The set N +
+ij = N +
+i
+∪ N +
+j
+is
+searched first to find nearest neighbours of the merged node
+efficiently (Prop. 3.3). The nodes in set N −
+ij need to update
+their nearest neighbours since their current nearest neigh-
+bour nodes i and j are getting contracted. Only the arcs
+to/from i and j are shown.
+bounded from above by
+bij := min
+p∈N +
+i
+⟨fi, fp⟩ + min
+q∈N +
+j
+⟨fj, fq⟩
+Proof. Since neighbours of i are computed by nearest
+neighbours search we have for all nodes p′ /∈ N +
+i
+⟨fi, fp′⟩ ≤ min
+p∈N +
+i
+⟨fi, fp⟩,
+and similarly for node j.
+Then by definition of v and
+Lemma 3.1 we obtain
+⟨fm, fv⟩ = ⟨fi, fv⟩ + ⟨fj, fv⟩
+≤ min
+p∈N +
+i
+⟨fi, fp⟩ + min
+q∈N +
+j
+⟨fj, fq⟩ .
+The above proposition gives an upper bound of feature sim-
+ilarity (i.e. edge cost) of merged node m with all nodes not
+in N +
+ij . Thus if a node in N +
+ij exceeds this upper bound it
+is more similar to m than all nodes not in N +
+ij . This allows
+to possibly skip recomputing the nearest neighbors of m in
+Alg. 2 (line 7).
+Lemma 3.4. If
+|{p ∈ N +
+ij : ⟨fm, fp⟩ ≥ bij}| ≥ k
+(8)
+then
+k-nearest
+neighbour
+of
+node
+m
+given
+by
+arg top-kv∈V \{i,j,m}⟨fm, fv⟩
+can
+be
+chosen
+as
+arg top-kp∈N +
+ij ⟨fm, fp⟩.
+Proof. Since the elements of N +
+ij already satisfy the bound
+bij from Prop. 3.3 and there are at least k many such el-
+ements, the k-nearest neighbours of node m can be taken
+from N +
+ij .
+
+Clustering Fully connected Graphs by Multicut
+Both of these approaches for efficiently updating the NN
+graph after contraction are used in Alg. 3. This algorithm
+can be used instead of lines 6 and 7 in Alg. 2 for improved
+performance. See Figure 2 for an illustration on nearest
+neighbour graph and edge contraction update.
+Algorithm 3: Incremental NN update
+Data: Contracting nodes i, j; Contracted node m; NN
+graph (V, A); Node features fi, ∀i ∈ V ; Num.
+of neighbours k;
+Result: Nearest neighbour arcs H to add in A
+// NNs of m by Prop. 3.3
+1 H = {(m, l)|l ∈ N +
+ij , ⟨fm, fl⟩ ≥ bij}
+// Keep at most k NN
+2 H = arg top-k(m,l)∈H⟨fm, fl⟩
+3 if H = ∅ then
+4
+H = {(m, r)|r = arg top-kl∈V \m⟨fm, fl⟩}
+5 for q ∈ N −
+ij \ {i, j} do
+// Check if m a NN of q
+6
+if ⟨fq, fm⟩ ≥ minl∈N +
+q ⟨fq, fl⟩ then
+7
+H = H ∪ (q, m)
+8
+else
+9
+H = H ∪ {(q, arg maxl∈V \q⟨fq, fl⟩)}
+3.2. Lazy Edge Contraction
+We further forego the need for nearest neighbours recom-
+putation after edge contraction by lifting the restriction of
+performing only greedy moves. This allows to maximally
+utilize the NN graph: the algorithm performs contractions,
+including non-greedy ones, until no contraction candidates
+are present in the NN graph. Specifically we do not per-
+form the exhaustive search in lines 4 and 9 and only return
+the nearest neighbours which are easily computable. The
+NN graph is repopulated as lazily as possible i.e. when no
+contraction candidates are left. In addition to being more
+efficient this strategy is reminiscent of the balanced edge
+contraction approach of (Kardoost & Keuper, 2018). The
+authors normalized the edge costs with cluster size of two
+end-points. These normalized edge costs were used to find
+the edge to contract. This strategy encouraged consecutive
+contractions to occur at different regions of the graph. As
+our lazy approach does not always make the nearest neigh-
+bours of the contracted node available thus contractions can
+only be done to nodes other than the contracted node. This
+also produces contractions in different regions.
+Lastly we also utilize efficient methods for approximate
+nearest neighbour search (Malkov & Yashunin, 2018) for
+populating the possibly large initial NN graph. For later
+nearest neighbour searches we still use exact methods as
+the search space is reduced due to previous contractions.
+3.3. Varying Affinity Strength
+Our basic edge costs computed by ⟨fi, fj⟩ for two features
+fi and fj have one fundamental limitation: Clusters will by
+default occupy whole quadrants. In other words, whenever
+two features have angle lower than 90◦ they are attractive
+and will prefer to be in the same cluster, see Figure 3. In
+order to let our formulation favor larger or smaller clusters,
+we modify our original similarity function s(·, ·) by adding
+an additional term indicated by α-variables:
+f i = [fi; αi],
+(9)
+s(f i, fj) = ⟨fi, fj⟩ ± αi · αj ,
+(10)
+where we choose positive sign for favoring larger clusters
+and negative for smaller clusters. In our experiments we
+will set αi = α with α > 0 a constant.
+We note that the contraction mechanism carries over di-
+rectly to our extended setting.
+Lemma 3.5. Aggregating features of the contracted node
+m by fm = f i + f j is equivalent to setting edge costs as
+per (4) on complete graph.
+Proof. Similar to the proof of Lemma 3.1 as follows
+s(f m, fl) = ⟨fm, fl⟩ ± αm · αl
+= ⟨fi + fj, fl⟩ ± (αi + αj) · αl
+= ⟨fi, fl⟩ ± αi · αl + ⟨fj, fl⟩ ± αj · αl
+= s(fi, f l) + s(f j, fl) .
+Large clusters:
+If we want to allow for larger clusters
+(corresponding to choosing + in (10)), we work directly on
+the extended feature set fi = [fi; αi] and use it in the NN
+graph.
+Small clusters:
+If we want to allow for smaller clusters
+(corresponding to choosing − in (10)), we must modify our
+algorithms slightly. In order to construct NN graphs we
+will use two sets of features: First, the query nodes will
+have their features defined by ˆfi = [fi, −αi] and second,
+the pre-existing nodes j ∈ V in the graph will keep the
+same features fj from (10). In order to search for nearest
+neighbors of node i in the graph V the modified similarity
+function (10) can be implemented by an inner product as
+s(f i, f j) = ⟨ ˆfi, f j⟩ .
+(11)
+4. Experiments
+We study the benefits of the multicut on complete
+graphs (3) and compare possible algorithms on the
+
+Clustering Fully connected Graphs by Multicut
+fd
+fe
+ff
+fg
+fh
+fa
+fb
+fc
+⟨fc, fb⟩ > 0
+⟨fc, fd⟩ > 0
+Figure 3: Illustration of edge costs between 8 nodes where
+feature vectors of each node i is in two-dimensional space
+i.e. fi ∈ R2. If we want each node to be a separate cluster
+then the edge costs measured by (5) are not suitable. This is
+because there will always be atleast two vectors with pos-
+itive costs preferring to be in the same cluster. Using an
+large enough positive value of α through (10) this issue can
+be resolved.
+tasks of ImageNet (Deng et al., 2009) clustering and
+Cityscapes (Cordts et al., 2016) panoptic segmentation.
+The algorithms are
+GAEC: The greedy additive edge contraction algorithm
+from (Keuper et al., 2015)(Alg. 1) is run on the com-
+plete graph where all edge costs are precomputed and
+then passed to the algorithm.
+RAMA: We also compare with the recent GPU-based mul-
+ticut solver of (Abbas & Swoboda, 2022). Similar to
+GAEC we run it on the complete graph. The solver
+uses dual optimization for better solution quality and
+also gives lower bounds to the multicut objective (1).
+As a drawback this solver cannot handle large in-
+stances due to high memory requirement of complete
+graphs. For running the solver we use an NVIDIA
+A40 GPU with 48GB of memory.
+DGAEC: Our Algorithm 2 which operates on node features
+and performs contractions according to Lemma 3.1.
+The nearest neighbour graph is updated by exhaustive
+search after edge contraction. The number of nearest
+neighbours k is set to 1.
+DGAECInc: Our Algorithm 2 which additionally makes
+use of Algorithm 3 for incrementally populating near-
+est neighbours after edge contraction. The value of k
+is set to 5.
+DLAEC: A variant of our DGAEC where non-greedy moves
+are also allowed as described in Sec. 3.2. The value of
+k is set to 5.
+DAppLAEC: Another variant of our DLAEC where initial
+nearest neighbours are computed by approximate near-
+est neighbour search method
+(Malkov & Yashunin,
+2018) through the implementation (Johnson et al.,
+2019).
+For all multicut algorithms on all datasets we set the value
+of affinity strength αi in (11) to 0.4, preferring small clus-
+ters. All CPU algorithms are run on an AMD 7502P CPU
+with a maximum of 8 threads to allow for faster NN search.
+4.1. ImageNet clustering
+We evaluate clustering of the ImageNet (Deng et al., 2009)
+validation set containing 50k images.
+Each image in
+the dataset acts as a node for our dense multicut for-
+mulation.
+The features of each image are computed
+by a ResNet50 (He et al., 2016) backbone trained by
+MoCov3 (Chen et al., 2021) in unsupervised fashion by a
+constrastive loss on the training split of ImageNet. The fea-
+tures have a dimension of 2048 and are normalized to have
+unit L2 norm. We create two problem instances containing
+5k and 50k images by considering 100 and all 1000 classes
+respectively.
+Clustering quality:
+Before comparing our algorithmic
+contributions we first test the efficacy of our dense
+multicut formulation by comparing its clustering re-
+sult with k-means (Lloyd, 1982) using the implemen-
+tation from (Pedregosa et al., 2011) and initialization
+of (Arthur & Vassilvitskii, 2007). Since k-means requires
+the number of clusters to be known beforehand we set it to
+the number of classes in the problem instance. For an ad-
+ditional comparison we also run k-means on the number of
+clusters given by our dense multicut algorithm. The quality
+of clustering results are evaluated using normalized mutual
+information (NMI) and adjusted mutual information (AMI)
+metrics (Vinh et al., 2010). The results are given in Table 1.
+We observe that although our formulation does not require
+the number of clusters to be specified, the results are on par
+with k-means. Additionally the value of affinity strength
+α does not need to be changed for different problem in-
+stances. As compared to k-means our algorithms are much
+faster especially on the larger instance. The RAMA solver
+of (Abbas & Swoboda, 2022) performs better than all other
+approaches on the smaller instance but runs out of mem-
+ory for the larger one. Lastly, our formulation creates more
+clusters than the number of classes. This is mainly due to
+presence of outliers in the feature space as the feature ex-
+tractor is trained without any groundtruth information.
+Algorithms comparison:
+We compare different algo-
+rithms for solving dense multicut problem (3) for imageNet
+clustering in Table 2. Firstly, we see that on the smaller
+
+Clustering Fully connected Graphs by Multicut
+Table 1: Comparison of clustering obtained by different
+methods on ImageNet validation set. t [s]: compute time
+in seconds, NMI: normalized mutual information, AMI: ad-
+justed mutual information, # clusters: number of clusters, †:
+out of GPU memory. For k-means the number of clusters
+was specified as input.
+Method
+t [s] ↓
+NMI ↑
+AMI ↑
+# clusters
+ImageNet-100 (|V | = 5k)
+k-means
+16
+0.42
+0.27
+100
+k-means
+32
+0.53
+0.26
+333
+RAMA
+0.9
+0.57
+0.29
+639
+DGAECInc
+42
+0.43
+0.22
+343
+DAppLAEC
+3.2
+0.47
+0.26
+333
+ImageNet-1000 (|V | = 50k)
+k-means
+701
+0.54
+0.2
+1000
+k-means
+1801
+0.61
+0.19
+2440
+RAMA
+†
+†
+†
+†
+DGAECInc
+2964
+0.49
+0.19
+2488
+DAppLAEC
+65
+0.56
+0.26
+2440
+instance the GPU based solver RAMA (Abbas & Swoboda,
+2022) gives the best performance. Secondly using incre-
+mental nearest neighbour search through Alg. 3 gives bet-
+ter run time than exhaustive search. Lastly our non-greedy
+algorithms give the best run time among all CPU-based al-
+gorithms although with slightly worse objectives.
+On the smaller instance, RAMA outperforms other algo-
+rithms in terms of the objective value (3) and also gives bet-
+ter clustering quality as compared to k-means. As a draw-
+back RAMA cannot handle large dense multicut instances.
+This shows multicut on complete graphs can be a suitable
+alternative to k-means. We speculate that algorithmic im-
+provements on top of our proposed algorithms will further
+improve clustering quality for large graphs.
+4.2. Panoptic segmentation
+We
+evaluate our method
+on
+the
+task
+of
+panoptic
+segmentation (Kirillov et al., 2019) on the Cityscapes
+dataset (Cordts et al., 2016). The panoptic segmentation
+task consists of assigning a class label to each pixel and
+partitioning different instances of classes with object cat-
+egories (e.g.
+car, person etc.).
+We focus on the task
+of partitioning for which the multicut formulation (1)
+has been used by (Kirillov et al., 2017; Abbas & Swoboda,
+2021).
+The latter work used a carefully crafted graph
+structure.
+Our dense multicut (3) formulation foregoes
+the need for finding a suitable graph structure. We use
+the pretrained Axial-ResNet50 (Wang et al., 2021) network
+Table 2: Comparison of algorithms for solving dense mul-
+ticut problem on two splits of Imagenet validation set. t [s]:
+compute time in seconds, Obj: objective value of cluster-
+ing (3), †: out of GPU memory, ⋆: no result within a 3 hour
+time limit.
+ImageNet-100
+ImageNet-1000
+Method
+t [s] ↓
+Obj ↓
+t [s] ↓
+Obj ↓
+GAEC
+24
+-6.84e5
+2605
+-9.353e7
+RAMA
+0.8
+-6.95e5
+†
+†
+DGAEC
+132
+-6.84e5
+⋆
+⋆
+DGAECInc
+42
+-6.84e5
+2934
+-9.353e7
+DLAEC
+5
+-6.83e5
+341
+-9.332e7
+DAppLAEC
+3.2
+-6.83e5
+65
+-9.332e7
+from (Yu et al., 2022), made available by (Weber et al.,
+2021) to compute the node features. Specifically, the net-
+work computes L2-normalized instance discriminative fea-
+tures in its intermediate stages which we use for our study
+without any training.
+For our evaluation we first compute semantic class predic-
+tions and then create a dense multicut instance for each se-
+mantic category with objects (i.e., car, person etc.). Such
+classes are also known as thing classes. The goal of the
+multicut problem is then to partition all nodes belonging
+to same semantic class to different objects. This strategy
+creates a total of 1631 dense multicut problem instances
+of varying sizes from 500 images of the Cityscapes valida-
+tion set. The largest problem instance contains around 43k
+nodes.
+Clustering quality:
+As a first point of comparison we
+check whether formulating a multicut problem on the com-
+plete graph by (3) is beneficial as compared to a hand-
+crafted sparse graph structure. We take the sparse graph
+structure from (Abbas & Swoboda, 2021) as a baseline.
+Their graph also includes long-range edges for dealing with
+occlusions leading to about 10 ·|V | edges in total. We com-
+pute the edge costs in this sparse graph in the same way as
+for our dense multicut formulation. For solving this multi-
+cut problem (1) we use Alg. 1.
+In Table 3 we compare the quality of clustering through
+the panoptic quality metric (Kirillov et al., 2019). We ob-
+serve that our dense multicut formulation performs better
+than multicut on the sparse handcrafted graph. This im-
+provement is significant for classes which can have many
+instances of the same class within an image (i.e. person,
+car) thus making the partitioning problem difficult. For
+classes with large objects (e.g. truck) having more edges
+does not help since the sparse graph can already capture
+
+Clustering Fully connected Graphs by Multicut
+Table
+3:
+Comparison
+of
+panoptic
+segmentation
+on Cityscapes dataset.
+Multicut on sparse graph
+of (Abbas & Swoboda, 2021) is computed by Alg. 1.
+For dense multicut we use the DAppLAEC algorithm.
+PQth: Average panoptic quality of all thing classes.
+Panoptic quality (%) ↑
+Category
+Sparse multicut
+Dense multicut
+Person
+40.0
+46.9
+Rider
+53.0
+54.4
+Car
+50.7
+60.5
+Truck
+52.7
+52.3
+Bus
+72.1
+71.1
+Train
+65.6
+62.9
+Motorcycle
+47.0
+46.8
+Bicycle
+45.7
+46.9
+PQth
+53.3
+55.2
+most inter-pixel relations. On average our dense multicut
+formulation gives better results than sparse multicut while
+alleviating the need for designing a graph structure.
+Algorithms comparison:
+We compare dense multicut al-
+gorithms for the panoptic segmentation task in terms of ob-
+jective value and run time. We were not able to run RAMA
+since the GPU could not store large graphs. The compar-
+ison of performance to the remaining algorithms averaged
+over all problem instances is given in Table 4. Moreover,
+in Figure 4 we compare performance of the algorithms on
+all large problem instances.
+In terms of run time, we see that our most naive algorithm
+DGAEC is slower than GAEC which directly operates on
+edge costs. Our other algorithms surpass GAEC reaching up
+to an order of magnitude run time improvement with lazy
+edge contractions and approximate initial nearest neigh-
+bours search. In terms of objective value we see slight im-
+provement by our lazy contraction algorithms as compared
+to the greedy ones.
+Sensitivity of affinity strength:
+In Table 5 we study the
+effect of changing the value of α from (10). We observe
+even better panoptic quality using a value of 0.3 as com-
+pared to our default of 0.4. As the edge costs lie in [−1, 1]
+due to L2-normalized node features, values of α close to
+0 or 1 gives more performance degradation. Last, we see
+further improvement if the value of α is set differently for
+each class. We refer to the Appendix for further results.
+−1.4 −1.2
+−1
+−0.8 −0.6 −0.4 −0.2
+0
+← Objective value (×108)
+10−1
+100
+101
+102
+103
+104
+← time [s]
+GAEC
+DGAEC
+DGAECInc
+DLAEC
+DAppLAEC
+Figure 4: Comparison of algorithms on large dense multi-
+cut instances (|V | ≥ 5000) from Cityscapes validation set.
+Overlaid bars mark the 0.25, 0.5 and 0.75-quantile.
+Table 4: Comparison of algorithms for solving dense mul-
+ticut problem on Cityscapes validation set. (t [s]): average
+compute times in seconds, (Obj): average objective value
+of clustering (3). The average is calculated over all problem
+instances.
+Method
+t [s] ↓
+Obj (×106) ↓
+GAEC
+10.0
+-6.338
+DGAEC
+84.1
+-6.338
+DGAECInc
+3.2
+-6.338
+DLAEC
+2.1
+-6.340
+DAppLAEC
+1.5
+-6.341
+Table 5: Results of panoptic segmentation via dense multi-
+cut with different values of attraction/repulsion strength α
+in (10). PQth: Avg. panoptic quality over all thing classes.
+α
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+PQth
+54.5
+55.8
+55.2
+55.0
+54.1
+52.0
+49.3
+
+Clustering Fully connected Graphs by Multicut
+5. Conclusion
+We have demonstrated that optimizing multicut on large
+complete graphs is possible when using factorized edge
+costs through inner products of features. We speculate that
+further algorithmic improvements are possible e.g. by per-
+forming dual optimization directly on the node features.
+As a potential theoretical advantage our approach sidesteps
+the need for learning graph structure. This offers a possibil-
+ity to embed it as a differentiable layer in neural networks,
+using e.g. the work (Vlastelica et al., 2019).
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+Clustering Fully connected Graphs by Multicut
+Appendix
+A. Influence of affinity strength
+On the Cityscapes dataset we compare panoptic quality on different object classes by varying the value of affinity strength
+α in (11). The results are given in Table 6. We observe that for classes contain many small objects large value of α is
+suitable whereas for classes with large objects small value of α is preferable. Although our default value of 0.4 already
+makes dense multicut outperform the baseline, further improvement is still possible e.g. by tuning α.
+Table 6: Comparison of panoptic segmentation on Cityscapes dataset for different values of affinity strength α (11). All
+results are computed using the DAppLAEC algorithm. Largest values in each row are highlighted with bold.
+Panoptic quality on varying values of α
+Category
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Person
+31.5
+38.1
+43.2
+46.9
+49.8
+52.6
+54.3
+55.0
+52.4
+Rider
+51.1
+53.0
+53.9
+54.5
+55.5
+55.4
+53.9
+51.0
+45.5
+Car
+45.6
+52.9
+57.8
+60.5
+63.3
+64.8
+64.1
+62.2
+57.8
+Truck
+54.1
+53.7
+52.7
+52.3
+49.0
+47.8
+45.4
+41.5
+34.7
+Bus
+75.1
+74.2
+73.5
+71.2
+69.3
+63.6
+58.5
+54.5
+47.3
+Train
+75.0
+74.9
+71.5
+62.9
+56.3
+51.7
+45.1
+40.4
+32.3
+Motorcycle
+45.5
+46.1
+48.0
+46.8
+48.7
+49.1
+47.8
+45.2
+39.8
+Bicycle
+38.1
+43.2
+45.6
+46.9
+47.8
+48.0
+46.9
+44.6
+40.4
+Average (PQth)
+52.0
+54.5
+55.8
+55.2
+55.0
+54.1
+52.0
+49.3
+43.8
+

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@@ -0,0 +1,1397 @@
+Determination of the top-quark mass from top-quark pair events with the matrix element method
+at next-to-leading order: Potential and prospects.
+Till Martini∗
+Fraunhofer Zentrum SIRIOS, Fraunhofer Institute for High-Speed Dynamics EMI, Berlin, Germany
+Turan Nuraliyev† and Peter Uwer‡
+Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, 12489 Berlin, Germany
+More than 25 years ago the matrix element method has been used in a pioneering work by D�0 to determine the
+top-quark mass from a handful of events. Since then the method has been matured into a powerful analysis tool.
+While the first applications were restricted to leading-order accuracy, in the meantime also the extension to next-
+to-leading order (NLO) accuracy has been studied. In this article we explore the potential of the matrix element
+method at NLO to determine the top-quark mass using events with pair-produced top quarks. We simulate a
+toy experiment by generating unweighted events with a fixed input mass and apply the matrix element method
+to construct an estimator for the top-quark mass. Two different setups are investigated: unweighted events
+obtained from the fixed-order cross section at NLO accuracy as well as events obtained using POWHEG matched
+to a parton shower. The latter lead to a more realistic simulation and allow to study the impact of higher-order
+corrections as well as the robustness of the approach. We find that the matrix element method in NLO accuracy
+leads to a significant reduction of the theoretical uncertainties compared to leading order. In view of the high
+luminosity phase of the LHC, this observation is especially relevant in analyses which are no longer dominated
+by statistical uncertainties.
+I.
+INTRODUCTION
+Regarding experimental as well as theoretical progress,
+hadronic top-quark pair production has evolved into one of
+the flagship processes at the LHC. This development is pro-
+pelled by the expectation of the top quark to play a prominent
+role in extensions of the Standard Model due to it being by far
+the heaviest of the elementary particles with a life time sig-
+nificantly shorter than the time scale of hadronization. The
+high production rate of top-quark pairs at the LHC as well
+as onward advances in experimental data taking enable for
+ever-decreasing statistical and systematic uncertainties in the
+recorded data. In order to make optimal use of this fact in ex-
+perimental analyses, the employed theoretical predictions are
+required to keep up in terms of uncertainties.
+The next-to-leading order QCD corrections for top-quark
+pair production have been calculated for the spin independent
+case more then 30 years ago [1–4]. Later, also the spin depen-
+dent cross sections were evaluated at NLO accuracy in QCD
+[5, 6]. In a series of ground breaking articles also the next-
+to-next-to-leading order QCD corrections were calculated [7–
+9]. Furthermore, beyond fixed order also the resummation of
+soft-gluon corrections has been studied in great detail ([10–
+17]). In addition to QCD corrections also weak and QED
+corrections have been calculated [18–22]. In summary, many
+detailed theoretical predictions for top-quark pair production
+are available. However, these might not be readily applicable
+in the experimental analysis. It is thus important to put more
+∗ Work on this article was conducted while employed at Humboldt-
+Universität
+zu
+Berlin,
+Institut
+für
+Physik,
+Berlin,
+Germany;
+Till.Martini@physik.hu-berlin.de
+† Turan.Nuraliyev@physik.hu-berlin.de
+‡ Peter.Uwer@physik.hu-berlin.de
+effort in improving the interface between experiment and the-
+ory to make optimal use of the increasing precision reached in
+both fields.
+Multivariate analysis methods like the matrix element
+method (MEM), turn out to be particularly useful in making
+optimal use of the theoretical predictions. The MEM requires
+the calculation of event weights in terms of differential cross
+sections and is thus often formulated at lower-order accuracy
+only. At leading order (LO), the MEM has been established
+as a powerful analysis tool for both signal searches as well as
+parameter inference by virtue of its optimal utilization of the
+information content of the available data. Typically, the im-
+pact of higher-order QCD corrections on theoretical predic-
+tions can be significant while often simultaneously decreasing
+the theoretical uncertainties. In the quest for accuracy and pre-
+cision to match experimental achievements, the MEM at next-
+to-leading order (NLO) represents a promising remedy. But
+when taking higher-order corrections into account, the calcu-
+lation of event weights constitutes a non-trivial task due to the
+intricate combination of virtual and real contributions to ob-
+tain meaningful finite results. The problem of extending the
+MEM beyond the Born approximation has been solved in the
+past by introducing modified jet algorithms on the one hand
+or sensible event definitions on the other hand ([23–25]). At
+the same time, the application of the MEM at NLO has been
+demonstrated for top-quark mass extraction from simulated
+single top-quark events ([23–25]) as well as anomalous cou-
+pling parameter determination from simulated Higgs boson
+events in association with a single top quark ([26]). Addi-
+tionally, the effects of a parton shower applied to simulated
+single top-quark data has been investigated with the MEM at
+NLO ([25]). In this work, we present the application of the
+MEM at NLO to top-quark pair production at the LHC. In
+contrast to the electroweak production mechanism of single
+top quarks studied before, top-quark pair production is QCD-
+induced at LO already with the two production channels of
+arXiv:2301.03280v1  [hep-ph]  9 Jan 2023
+
+2
+quark-antiquark annihilation and gluon-gluon fusion consti-
+tuting the dominant source of top quarks at the LHC. Given
+the aforementioned prominent role of top-quark pair produc-
+tion in both experimental as well as theoretical advances at
+the LHC, it represents an ideal example to study higher-order
+effects within the MEM. Furthermore, in view of the ongo-
+ing progress in top-quark mass measurements, the MEM at
+NLO accuracy could be an interesting alternative to existing
+approaches.
+The paper is structured as follows. In section II the NLO
+QCD calculation of the differential cross section for top-quark
+pair production with the phase space slicing method and the
+subsequent generation of unweighted events are briefly re-
+viewed. Section III focuses on the application of the MEM to
+the generated events. To study parton shower effects, events
+generated with POWHEG+Pythia [27–31] are also analysed.
+The conclusions are presented in section IV.
+II.
+TOP-QUARK PAIR PRODUCTION AT THE LHC
+A.
+Implementing the NLO prediction with the phase space
+slicing method
+The MEM at NLO as presented in [23–25] requires the
+cross-section calculation at NLO to be carried out using the
+phase space slicing method [32]. The respective calculation
+is available in the literature [5]. Thus, in this section we only
+give a brief review of the important aspects of the calcula-
+tion and present the validation for the choice of the slicing pa-
+rameter. In the phase space slicing method, the cross-section
+prediction at NLO accuracy dσNLO is formed of two contri-
+butions: First, the so-called hard part dσHard is just the ma-
+trix element for the real corrections evaluated for phase space
+points where all partons are resolved, that is the additional
+parton is neither collinear to the incoming partons nor soft.
+Second, a Born-like part is comprised of the Born contribution
+dσLO, the virtual corrections dσvirtual (taken from Ref. [33])
+as well as the so-called soft and collinear parts dσsoft/coll.
+stemming from approximated real corrections integrated over
+phase space regions in which the additional parton is unre-
+solved. The separation of the phase space for the real cor-
+rections into resolved and unresolved regions is mediated by
+the so-called slicing parameter xmin which acts as a scale to
+separate the two. In the unresolved regions, well-known fac-
+torization properties of QCD real corrections can be employed
+allowing to analytically integrate over the additional radiation
+in the singular limits in an approximate way thereby reducing
+the respective phase space to Born-like kinematics. The diver-
+gences of these integrations can be regularized within dimen-
+sional regularization leading to poles in the dimensional shift
+away from four space-time dimensions. The outcome can be
+combined with the virtual contributions to cancel the respec-
+tive poles from the loop integration and yield finite results ac-
+cording to the Kinoshita-Lee-Nauenberg theorem ([34, 35]).
+Since the real corrections are approximated in the unresolved
+(singular) regions, the result is only accurate up to deviations
+10−6
+10−5
+10−4
+10−3
+xmin
+703
+704
+705
+706
+707
+708
+709
+σNLO[pb]
+reference
+σNLO
+FIG. 1. Phase space slicing parameter (in-)dependence of the total
+cross section predicted at NLO accuracy. The red line shows the
+reference value taken from HATHOR [36].
+0
+10
+20
+30
+40
+dσNLO
+dk⊥
+1
+[pb GeV−1]
+xmin = 0.0002
+xmin = 0.0001
+xmin = 0.00005
+0
+100
+200
+300
+400
+500
+k⊥
+1 [GeV]
+−2σ
+−σ
+σ
+2σ
+pull
+xmin = 0.0002 vs xmin = 0.0001
+xmin = 0.0001 vs xmin = 0.00005
+FIG. 2. Phase space slicing parameter (in-)dependence of the top-
+quark transverse momentum predicted at NLO accuracy.
+proportional to the slicing parameter xmin:
+dσNLO = dσHard + dσLO + dσvirtual + dσsoft/coll. + O(xmin) .
+(1)
+Additionally, the separation of the real phase space in terms of
+the slicing parameter introduces logarithmic dependencies of
+the hard and soft/collinear contributions on xmin which cancel
+in the sum. However, when numerically integrating over the
+finite hard contribution, these logarithms can lead to numeri-
+cal instabilities if xmin is chosen too small. Hence, the value
+of xmin has to be chosen as a compromise between numeri-
+cal stability and the demand that the deviation in Eq. (1) is
+negligible compared to the statistical uncertainties of the total
+cross section as well as distributions calculated at NLO accu-
+racy. Fig. 1 shows NLO predictions for the total cross section
+of top-quark pair production for different values of the slicing
+parameter xmin. The total cross section as the sum of Born,
+virtual and real contributions in Fig. 1 is indeed finite. How-
+ever, it shows a systematic deviation from the reference value
+taken from HATHOR [36] for values xmin ⪆ 2 × 10−3 while for
+values xmin ⪅ 5 × 10−6 numerical instabilities dominate. Ac-
+cordingly, a value of xmin = 10−4 is chosen. As an example
+of a differential distribution the top-quark transverse momen-
+
+3
+0
+10
+20
+30
+40
+dσ
+dk⊥
+1 [pb GeV−1]
+dσNLO
+dσLO
+0
+100
+200
+300
+400
+500
+k⊥
+1 [GeV]
+1.0
+1.5
+2.0
+dσNLO
+dσLO
+0
+10
+20
+30
+40
+50
+dσ
+dη1 [pb]
+dσNLO
+dσLO
+−4
+−2
+0
+2
+4
+η1
+1.0
+1.5
+2.0
+dσNLO
+dσLO
+FIG. 3.
+Differential distributions together with the respective k-
+factors.
+tum calculated at NLO accuracy is shown for three different
+choices of xmin in Fig. 2. In the lower plot we show for dif-
+ferent choices of xmin the differences in units of the statistical
+uncertainties. We conclude that all three choices lead to coher-
+ent predictions justifying the choice xmin = 10−4. In addition
+to the top-quark transverse momentum this has been checked
+also for the top-quark energy distribution and the rapidity dis-
+tribution. Furthermore, the distributions calculated here have
+been cross checked with results from madgraph5 aMC@NLO
+[37]. The comparison is shown in appendix A, Fig. 9 and
+Fig. 10. The impact of the NLO corrections on kinematic dis-
+tributions is displayed in Fig. 3 where NLO and LO predic-
+tions for kinematic distributions are compared and their ratios
+(the k-factor) are shown at the bottom of the plots. Results
+for further distributions are shown in Fig. 11 in appendix A.
+As can be seen from the rather constant k-factors, the NLO
+corrections only mildly affect the shapes of the kinematic dis-
+tributions. However, the NLO corrections lead to a significant
+increase of the cross sections by a factor of roughly 1.5. In
+Fig. 4 the impact of variations of the factorization scale µF
+and renormalization scale µR by a factor of 2 as a means to es-
+timate the effect of un-calculated higher orders are illustrated
+for the shapes of two representative kinematic distributions of
+the top quark. For moderate energy scales, one observes a
+significant reduction of the impact of the scale variation.
+0
+200
+400
+600
+800
+1000
+k⊥
+1 [GeV]
+10−6
+10−5
+10−4
+10−3
+10−2
+1
+σ
+dσ
+dk⊥
+1
+[GeV−1]
+LO
+NLO
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+y
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+1
+σ
+dσ
+dy
+LO
+NLO
+FIG. 4. Effect of scale variations on the shapes of kinematic distri-
+butions of the top quark.
+B.
+Unweighted event generation
+From the calculation of the cross section at NLO accu-
+racy outlined in the previous section, event weights can be
+calculated which can be used to generate unweighted events
+which are distributed according to the NLO cross section. As
+described in Ref. [25], a sensible event definition is manda-
+tory for obtaining meaningful event weights at NLO accuracy.
+In particular, the event definition must not fix the invariant
+masses or the overall transverse momentum of the final-state
+objects. For top-quark pair production, we define events ⃗x
+in terms of the transverse momentum k⊥
+1 , azimuthal angle φ1
+and pseudo rapidity η1 of the top quark as well as the pseudo
+rapidity of the antitop quark η2:
+⃗x = (k⊥
+1 ,φ1,η1,η2) .
+(2)
+The two-particle Born phase space as well as the three-particle
+phase space for the real radiation can be parameterized in
+terms of these variables
+dR2 =
+k⊥
+1
+3 coshη1 coshη2
+8π2 E1 E2 shad
+dk⊥
+1 dφ1 dη1 dη2 ,
+(3)
+dR3 =
+k⊥
+1
+2 k⊥
+2 k⊥
+3
+2 coshη1 coshη2 coshη3
+128π5 E1 E2 E3 shad
+× dk⊥
+1 dφ1 dη1 dη2 dk⊥
+3 dφ3 dη3 ,
+(4)
+
+4
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+1
+σNLO
+dσNLO
+dk⊥
+1
+[pb GeV−1]
+results
+reference
+0
+100
+200
+300
+400
+500
+k⊥
+1 [GeV]
+−2σ
+−σ
+σ
+2σ
+pull
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+0.07
+1
+σNLO
+dσNLO
+dη1
+[pb]
+results
+reference
+−10.0
+−7.5
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+η1
+−2σ
+−σ
+σ
+2σ
+pull
+FIG. 5. Validation of the event generation: Comparison of differen-
+tial distributions of the top quark obtained from unweighted events
+with results from madgraph5 aMC@NLO.
+where Ei denotes the energy of particle i and shad is the
+hadronic center-of-mass energy squared. The additional ra-
+diation occurring in the real corrections is parametrized by
+the transverse momentum k⊥
+3 , the azimuthal angle φ3 and the
+pseudo rapidity η3 of the radiated parton. These parametriza-
+tions allow together with Eq. (1) to calculate the event weight
+at NLO accuracy for each event ⃗x using
+d4σNLO
+dk⊥
+1 dφ1 dη1 dη2
+=
+d4σLO
+dk⊥
+1 dφ1 dη1 dη2
++
+�
+d7σHard
+dk⊥
+1 dφ1 dη1 dη2 dk⊥
+3 dφ3 dη3
+dk⊥
+3 dφ3 dη3
++
+d4σvirtual
+dk⊥
+1 dφ1 dη1 dη2
++
+d4σsoft/collinear
+dk⊥
+1 dφ1 dη1 dη2
+.
+(5)
+The weights calculated in this way can also be used to gener-
+ate unweighted events with, e.g., the von-Neumann acception-
+rejection method ([38]). Fig. 5 shows the distribution of the
+unweighted events compared to kinematic distributions ob-
+tained with the madgraph5 aMC@NLO code [37]. The events
+obtained from the event weights defined in Eq. (5) are within
+the uncertainties in perfect agreement with the predictions ob-
+tained using madgraph5 aMC@NLO. In appendix A, Fig. 12
+we show in addition the calculation of the Mt¯t- and the φ1-
+distribution with the same perfect agreement. The compari-
+son of the generated unweighted events with the results from
+madgraph5 aMC@NLO also serves as a further validation for
+the choice of the slicing parameter.
+III.
+APPLICATION: DETERMINATION OF THE
+TOP-QUARK MASS USING THE MEM AT NLO
+The event weights defined in Eq. (5) can be used in the
+MEM to calculate the likelihood at NLO accuracy for a given
+sample of N events {⃗xi}, i = 1,...,N:
+L�{⃗xi} | mt
+� =
+1
+(σNLO(mt))N
+N
+�
+i=1
+d4σNLO(mt)
+dk⊥
+1 dφ1 dη1 dη2
+������⃗x=⃗xi
+(6)
+where the dependence of the total and differential cross
+sections on the value of the top-quark mass is high-
+lighted—exemplarily for generic model parameters. Here, the
+so-called transfer functions, parametrizing the probability of
+measuring a certain signal in the detector given a particular
+partonic configuration, are set to delta functions. The trans-
+fer functions account for particle decays, additional radiation
+as well as detector effects. Thus, this choice for the transfer
+functions corresponds to the assumption of a perfect detector
+which allows a perfect unfolding from the detector signals to
+partonic variables. While for variables related to angles, set-
+ting the transfer function to delta function may give a reason-
+able approximation, this is not necessarily true in case of vari-
+ables sensitive to energies. In future applications non-trivial
+transfer functions should thus be incorporated. This may be
+done using invertible neural networks trained to a full simula-
+tion as discussed in great detail in Ref. [39]. This is however
+beyond the scope of this work which focuses on exploring
+the potential of the method for top-quark mass measurements.
+Maximizing the likelihood with respect to the parameter mt
+yields an estimator for the top-quark mass ˆmt:
+L�{⃗xi} | ˆmt
+� = max
+mt
+�L�{⃗xi} | mt
+�� .
+(7)
+Because the event weights in Eq. (6) are normalized to yield
+probabilities, the MEM is only sensitive to the shapes of kine-
+matic distributions but not to the total number of events in the
+sample. To also benefit from the information of the total event
+number the so-called extended likelihood can be used. The
+extended likelihood is obtained from the likelihood in Eq. (7)
+by multiplying with the Poisson probability for observing N
+events when the expected number of events is given by the
+total cross section times the integrated luminosity Lint of the
+collider:
+Lext
+�{⃗xi} | mt
+� = (σNLO(mt) Lint)N
+N!
+e−σNLO(mt) Lint L�{⃗xi} | mt
+�.
+(8)
+In Fig. 6 we show the likelihood obtained analysing 9900
+unweighted top-quark pair events distributed according to the
+NLO prediction. Likelihood (upper plot) as well as the ex-
+tended likelihood (lower plot) have been studied. The green
+curves correspond to likelihoods calculated at NLO accuracy
+
+5
+160
+165
+170
+175
+180
+185
+190
+m [GeV]
+−50
+−25
+0
+25
+50
+75
+100
+125
+150
+− log
+�
+L(m)
+Lmax
+�
+mtrue = 173.2 GeV, 9900 analysed events
+LO prediction:
+ˆm2µ0
+0.5µ0 = 169.77 ± 1.18+2.21
+−2.66GeV
+NLO prediction:
+ˆm2µ0
+0.5µ0 = 173.65 ± 1.20+0.30
++0.17GeV
+140
+150
+160
+170
+180
+190
+m [GeV]
+−100
+0
+100
+200
+300
+400
+500
+600
+700
+800
+− log
+�
+Lext(m)
+Lext,max
+�
+mtrue = 173.2 GeV, 9900 analysed events
+LO prediction:
+ˆm2µ0
+0.5µ0 = 160.22 ± 0.34−6.97
++7.86GeV
+NLO prediction:
+ˆm2µ0
+0.5µ0 = 173.68 ± 0.36−4.11
++3.69GeV
+FIG. 6.
+Analysis of unweighted events following the fixed-order
+NLO prediction with (extended) likelihoods calculated at LO and
+NLO accuracy.
+ˆmt ±∆stat
+∆
+2µ0
+sys
+∆
+µ0/2
+sys
+[GeV]
+likelihood
+LO prediction
+NLO prediction
+L 169.77±1.18+2.21
+−2.66 173.65±1.20+0.30
++0.17
+Lext 160.22±0.34−6.97
++7.86 173.68±0.36−4.11
++3.69
+TABLE I. Extracted values for the estimator of the top-quark mass
+from 9900 unweighted events following the fixed-order NLO predic-
+tion.
+using different choices for the factorization and renormaliza-
+tion scale. The orange curves are obtained using only LO
+predictions again for different scale settings in the likelihood
+calculation. The analysed events are generated for an input
+value of the top-quark mass of mtrue = 173.2 GeV and the scale
+choice µF = µR = µ0 = mt. The extracted values for the estima-
+tor of the top-quark mass together with statistical and system-
+atic uncertainties are summarized in Tab. I. The estimators ˆmt
+are determined from the minima of the parabolas fitted to the
+negative logarithms of the likelihood functions while the sta-
+tistical uncertainties ∆stat are estimated from their widths. The
+systematic uncertainties ∆2µ0
+sys , ∆µ0/2
+sys
+are estimated by varying
+the scale by a factor 2 around µ0. As can be seen from Fig. 6
+and Tab. I, both the NLO and the LO analyses have similar
+statistical uncertainties of about 1.2 GeV and 0.35 GeV de-
+pending on whether the likelihood or the extended likelihood
+is employed. As expected, the statistical uncertainties are to
+good approximation independent from the perturbative order
+of the theoretical predictions of the cross sections. Taking the
+statistical uncertainties into account, the extracted estimators
+from the NLO analyses are in perfect agreement with the input
+value. For the likelihood as well as for the extended likelihood
+the NLO differential cross section matches the probability dis-
+tribution underlying the event sample thus leading to an unbi-
+ased estimator. Obviously, taking into account the information
+on the total number of events via the extended likelihood leads
+to a reduction of the statistical uncertainties as additional in-
+formation contained in the event sample is used. Since the
+cross section shows a much stronger residual scale depen-
+dence than the normalized distributions, the extended likeli-
+hood leads however to a significantly larger systematic un-
+certainty due to uncalculated higher order corrections. In ad-
+dition, the uncertainty of the luminosity measurement which
+is not taken into account in the extended likelihood analysis
+leads to an additional uncertainty outweighing the gain in the
+reduced statistical uncertainty.
+The estimators from the LO analyses on the other hand
+show a bias of 2.9×∆stat and 38×∆stat depending on whether
+the likelihood or the extended likelihood is used. It should
+be emphasized that the occurrence of a bias per se does not
+rule out the application of the MEM. It is well known, that
+the MEM typically leads to a bias if the probability distribu-
+tion used in the evaluation of the likelihood does not match
+the distribution underlying the event sample. However, via
+a calibration procedure it is possible to compensate the bias
+and obtain an unbiased determination. Since the calibration
+can introduce additional uncertainties the preferred situation
+is that the probability distribution used in the likelihood de-
+termination matches the probability distribution of the event
+sample as best as possible thus reducing the need of addi-
+tional calibration. As shown in section II, the NLO correc-
+tions dominantly alter the normalization of the kinematic dis-
+tributions rather than their shape. Accordingly, the analysis
+employing extended likelihoods which is sensitive to the total
+cross section shows thus a much stronger separation between
+the results obtained from the NLO and LO predictions.
+Significant improvement from taking NLO corrections into
+account can be seen in their impact on the theoretical uncer-
+tainties: In the NLO analyses the theoretical uncertainties due
+to uncalculated higher order corrections are roughly halved
+with respect to the LO analyses.
+In order to further study the robustness of the approach
+and having a more realistic simulation, unweighted events ob-
+tained from a parton shower simulation matched to the NLO
+calculation can be used. The parton shower resums certain
+logarithmic corrections to all orders on top of the fixed-order
+NLO parton level calculation. Since these additional correc-
+tions present in the event sample are not accounted for in the
+fixed-order-only likelihood calculation based on Eq. (5), there
+is a mismatch between the underlying probability distribution
+of the generated events and the basis of the likelihood calcula-
+tion (Eq. (5)). As seen before in case of the LO analysis, this
+
+6
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+1
+σNLO
+dσNLO
+dk⊥
+1
+[GeV−1]
+POWHEG events
+fixed-order NLO
+0
+100
+200
+300
+400
+500
+k⊥
+1 [GeV]
+0
+1
+2
+POWHEG
+fixed-order NLO
+0.00
+0.02
+0.04
+0.06
+0.08
+1
+σNLO
+dσNLO
+dη1
+POWHEG events
+fixed-order NLO
+−10.0
+−7.5
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+η1
+0
+1
+2
+POWHEG
+fixed-order NLO
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+1
+σNLO
+dσNLO
+dy
+POWHEG events
+fixed-order NLO
+−4
+−2
+0
+2
+4
+y
+0
+1
+2
+POWHEG
+fixed-order NLO
+FIG. 7. Impact of the parton shower on the kinematic distributions
+of the top quark.
+mismatch can cause a systematic bias in the extracted estima-
+tor.
+Fig.
+7
+shows
+the
+distributions
+obtained
+using
+POWHEG+Pythia
+[27–31]
+to
+generate
+about
+the
+same
+number of events as in the case of the fixed-order analysis.
+The parton shower only mildly affects the kinematic distribu-
+tions relevant for the event definition. Further distributions
+supporting this observation are shown in Fig. 13 in the ap-
+pendix A. Apart from minor differences in the k⊥
+1 distribution
+at low k⊥
+1 , a small difference is visible for k⊥
+1 > 300 GeV,
+where the POWHEG+Pythia events lead to a slightly harder
+distribution than the events generated from the fixed order
+NLO cross section.
+160
+165
+170
+175
+180
+185
+190
+m [GeV]
+0
+100
+200
+300
+400
+− log
+�
+L(m)
+Lmax
+�
+mtrue = 173.2 GeV, 9232 analysed events
+LO prediction:
+ˆm2µ0
+0.5µ0 = 173.88 ± 1.22+2.13
+−2.57GeV
+NLO prediction:
+ˆm2µ0
+0.5µ0 = 177.93 ± 1.24+0.22
++0.38GeV
+FIG. 8. Analysis of 9232 POWHEG+Pythia events with fixed-order
+likelihoods calculated at LO and NLO accuracy.
+ˆmt ±∆stat
+∆
+2µ0
+sys
+∆
+µ0/2
+sys
+[GeV]
+likelihood
+LO prediction
+NLO prediction
+L 173.88±1.22+2.13
+−2.57 177.93±1.24+0.22
++0.38
+TABLE II. Extracted values for the estimator of the top-quark mass
+from unweighted POWHEG+Pythia events following the NLO predic-
+tion matched to a parton shower.
+The result of the likelihood analysis using LO and NLO
+cross section predictions is shown in Fig. 8 and summarized
+in Tab. II. We do not study the extended likelihood, since the
+extended likelihood leads to much larger systematic uncer-
+tainties. Again the statistical uncertainties are very similar for
+the LO and NLO analysis, while the systematic uncertainty is
+significantly reduced when using NLO predictions. In both
+cases we observe a shift of about 4 GeV compared to the re-
+sults based on the event sample generated from the fixed-order
+NLO predictions. The large shift shows the high sensitivity of
+the MEM with respect to tiny changes in the distributions. In
+a mass determination from events registered at the LHC this
+shift must be taken into account via a calibration procedure.
+It is remarkable that the shift is, taking the uncertainties into
+account, independent from the perturbative order of the em-
+ployed likelihood calculation. This is similar to what has been
+observed in Refs. [24, 25]. The LO likelihood analysis repro-
+duces the true mass value used in the POWHEG+Pythia analy-
+sis. However, this ist most likely accidental and due to the fact
+that the LO fixed-order results undershoots the true mass value
+by about 4 GeV which is compensated by the aforementioned
+shift.
+IV.
+CONCLUSION
+In this work the MEM at NLO is applied to top-quark
+pair production at the LHC. To investigate the potential of
+the matrix element method to measure the top-quark mass,
+the MEM at NLO is applied to pseudo-data: unweighted
+
+7
+events generated from the fixed-order NLO cross section
+as well as events obtained using POWHEG+Pythia incorpo-
+rating the parton shower effects. Using pseudo-data based
+on POWHEG+Pythia allows to study the effect of the parton
+shower and gives a more realistic simulation. Including the
+NLO corrections in the likelihood calculation leads to a signif-
+icant reduction of the theoretical uncertainties of the extracted
+top-quark mass, while the statistical uncertainties remain al-
+most unchanged compared to the LO analysis. We stress that
+the uncertainties due to scale variation cannot be reduced by a
+calibration. The reduction of the uncertainties associated with
+the scale variation when going from LO to NLO thus presents
+an important improvement and a strong argument in favour of
+the the MEM at NLO accuracy.
+Another important observation is the fact that the extended
+likelihood yields a significant improvement in terms of the sta-
+tistical uncertainties. However, in practical applications this
+gain in precision is completely outweighed by the theoretical
+uncertainties of the number of expected events. This can be
+understood from the fact that, much as the NLO corrections
+(see Fig. 3), the scale variations do not dramatically change
+the shape of the kinematic distributions but mostly their nor-
+malization (see Fig. 4) thereby making the extended likeli-
+hood analyses more sensitive to their effect. Additionally, em-
+ploying the extended likelihood requires precise knowledge of
+the integrated luminosity of the LHC. The dependence on this
+parameter introduces an additional source of systematic un-
+certainty. This has to be taken into account for future experi-
+mental applications of the MEM with realistic event numbers
+for abundantly produced top-quark pairs at the LHC which
+will most likely be dominated by systematic uncertainties. As
+has already been stated before ([23–26]), for parameter infer-
+ence with the MEM it is mandatory to perform the likelihood
+calculation at least at NLO accuracy in order to properly fix
+the renormalization scheme of the extracted parameter.
+The application of the MEM at NLO to top-quark pair
+events at the LHC can offer an alternative approach to deter-
+mine the top-quark mass with high accuracy. As has been
+demonstrated in this work, already for a few ten thousand
+events the precision of the analysis becomes dominated by
+systematic uncertainties. As the LHC produces millions of
+top-quark pairs, the analysis could be performed with a rather
+small fraction of cherry-picked events allowing to minimize
+the overall systematic uncertainty. The results obtained in this
+article suggest that top-quark mass determination with an un-
+certainty below 1 GeV could be feasible. Of course, for a re-
+alistic application of the MEM to experimental data, transfer
+functions accounting for decays, additional radiation and de-
+tector effects have to be considered. In addition, as the analy-
+sis based on the events including parton shower effects shows,
+a further calibration is required.
+ACKNOWLEDGMENTS
+This work was supported in part by the Bundesministerium
+für Bildung and Forschung under contract 05H18KHCA1.
+0
+10
+20
+30
+40
+50
+60
+dσNLO
+dE1
+[pb GeV−1]
+result
+reference
+0
+200
+400
+600
+800
+1000
+E1 [GeV]
+−2σ
+−σ
+σ
+2σ
+pull
+0
+10
+20
+30
+40
+50
+dσNLO
+dη1
+[pb]
+result
+reference
+−10.0
+−7.5
+−5.0
+−2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+η1
+−2σ
+−σ
+σ
+2σ
+pull
+FIG. 9. Validation of the implementation: Comparison of differential
+distributions of the top quark obtained in this work with results from
+madgraph5 aMC@NLO.
+Appendix A: Additional results on distributions used for the
+validation
+In this appendix we show further cross checks used for the
+validation of the implementation. Fig. 9 shows comparisons
+of NLO predictions for differential distributions calculated
+in this work with distributions obtained from madgraph5
+aMC@NLO [37] which is based on the dipole subtraction
+method [40, 41]. The pull distributions in the bottom plots
+of Fig. 9 and Fig. 10 illustrate the agreement between both
+implementations within statistical uncertainties. This compar-
+isons serve as a further validation for the choice of the slicing
+parameter. Fig. 11 shows the NLO corrections (upper part)
+together with the k-factors (lower part) for the Mt¯t and the φ1-
+distribution. Similar to what is shown in Fig. 3 again a flat k-
+factor is observed. As a check of the event generation and the
+unweighting procedure Fig. 12 shows distributions calculated
+from the generated unweighted events compared with a calcu-
+lation using madgraph5 aMC@NLO [37]. Similar to Fig. 7 we
+show in Fig. 13 for further distributions the comparison of dis-
+tributions obtained at fixed-order NLO accuracy with results
+using POWHEG+Pythia.
+
+8
+0
+5
+10
+15
+dσNLO
+dφ1
+[pb]
+result
+reference
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+φ1
+−2σ
+−σ
+σ
+2σ
+pull
+0
+10
+20
+30
+40
+dσNLO
+dk⊥
+1
+[pb GeV−1]
+result
+reference
+0
+100
+200
+300
+400
+500
+k⊥
+1 [GeV]
+−2σ
+−σ
+σ
+2σ
+pull
+FIG. 10. Same as Fig. 9 but for the φ1- and the k⊥
+1 -distribution.
+0
+10
+20
+30
+40
+dσ
+dMt¯t [pb GeV−1]
+dσNLO
+dσLO
+300
+400
+500
+600
+700
+800
+900
+1000
+Mt¯t [GeV]
+1.0
+1.5
+2.0
+dσNLO
+dσLO
+0
+5
+10
+15
+20
+dσ
+dφ1 [pb]
+dσNLO
+dσLO
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+φ1
+1.0
+1.5
+2.0
+dσNLO
+dσLO
+FIG. 11. Same as Fig. 3 but for the Mt¯t and the φ1-distribution.
+
+9
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+1
+σNLO
+dσNLO
+dMt¯t
+[pb GeV−1]
+results
+reference
+300
+400
+500
+600
+700
+800
+900
+1000
+Mt¯t[GeV]
+−2σ
+−σ
+σ
+2σ
+pull
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+1
+σNLO
+dσNLO
+dφ1
+[pb]
+results
+reference
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+φ1
+−2σ
+−σ
+σ
+2σ
+pull
+FIG. 12. Same as Fig. 5 but for the Mt¯t- and the φ1-distribution.
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+1
+σNLO
+dσNLO
+dE1
+[GeV−1]
+POWHEG events
+fixed-order NLO
+0
+200
+400
+600
+800
+1000
+E1 [GeV]
+0
+1
+2
+POWHEG
+fixed-order NLO
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+1
+σNLO
+dσNLO
+dMt¯t
+[GeV−1]
+POWHEG events
+fixed-order NLO
+300
+400
+500
+600
+700
+800
+900
+1000
+Mt¯t [GeV]
+0
+1
+2
+POWHEG
+fixed-order NLO
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+1
+σNLO
+dσNLO
+dφ1
+POWHEG events
+fixed-order NLO
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+φ1
+0
+1
+2
+POWHEG
+fixed-order NLO
+FIG. 13. Same as Fig. 7 but for the energy, Mt¯t- and φ1-distribution.
+
+10
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+Householder, G. E. Forsythe, and H. H. Germond (US Gov-
+ernment Printing Office, Washington, DC, 1951) Chap. 13, pp.
+36–38.
+[39] A. Butter, T. Heimel, T. Martini, S. Peitzsch,
+and T. Plehn,
+(2022), arXiv:2210.00019 [hep-ph].
+[40] S. Catani and M. Seymour, Nucl.Phys. B485, 291 (1997),
+arXiv:hep-ph/9605323 [hep-ph].
+[41] S. Catani, S. Dittmaier, M. H. Seymour,
+and Z. Trocsanyi,
+Nucl.Phys. B627, 189 (2002), arXiv:hep-ph/0201036 [hep-ph].
+

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@@ -0,0 +1,1086 @@
+arXiv:2301.03051v1  [math.RA]  8 Jan 2023
+FUNCTORS BETWEEN REPRESENTATION CATEGORIES.
+UNIVERSAL MODULES
+A. L. AGORE
+Abstract. Let g and h be two Lie algebras with h finite dimensional and consider
+A = A(h, g) to be the corresponding universal algebra as introduced in [4]. Given an
+A-module U and a Lie h-module V we show that U ⊗ V can be naturally endowed
+with a Lie g-module structure. This gives rise to a functor between the category of Lie
+h-modules and the category of Lie g-modules and, respectively, to a functor between
+the category of A-modules and the category of Lie g-modules.
+Under some finite
+dimensionality assumptions, we prove that the two functors admit left adjoints which
+leads to the construction of universal A-modules and universal Lie h-modules as the
+representation theoretic counterparts of Manin-Tambara’s universal coacting objects
+[11, 16].
+Introduction
+The universal coacting bialgebra/Hopf algebra on a finite dimensional (graded) asso-
+ciative algebra originates in the work of Yu. I. Manin ([11]). The importance of this
+construction became obvious mostly due to its interaction with non-commutative geom-
+etry where it is seen as some sort of symmetry group (see [13] for more details on this
+view point). The non-graded version of this construction appeared a few years later in a
+paper by D. Tambara ([16]). However, as remarked in [16], the universal coacting bialge-
+bra is in fact the dual of the so-called universal measuring bialgebra introduced by M.E.
+Sweedler in [15]. We should note that, unlike Manin-Tambara’s construction, Sweedler’s
+universal measuring bialgebra/Hopf algebra exists even in the infinite-dimensional case.
+In recent years, universal (co)acting objects have been considered in various settings
+and for different purposes. For instance, [8] extends Sweedler’s construction to monoids
+in a braided monoidal category. On the other hand, the Manin-Tambara construction
+was introduced in the setting of Poisson algebras ([2]), finite index-subfactors ([6]), su-
+perpotential algebras ([7]), polynomial algebras ([14]), bialgebroids ([5]) or Lie/Leibniz
+algebras ([4]). The corresponding universal coacting bialgebras/Hopf algebras, which in
+certain cases carry some extra structure (e.g. a Poisson Hopf algebra structure as in [2]),
+seem to play a prominent role in solving other seemingly unrelated problems such as
+the classification of gradings on various kinds of algebras ([4, 12]), the description of the
+automorphisms group of certain algebraic structures ([4]) and even in quantum Galois
+2010 Mathematics Subject Classification. 16D90, 16T05, 17A32, 17B10.
+Key words and phrases. universal module.
+This work was supported by a grant of Romanian Ministry of Research, Innovation and Digitization,
+CNCS/CCCDI – UEFISCDI, project number PN-III-P4-ID-PCE-2020-0458, within PNCDI III.
+1
+
+2
+A. L. AGORE
+theory ([6]). Another related universal (co)acting construction was considered in [3] as
+the Hopf algebraic analogue of the universal group of a grading and its connections to
+the problem of classifying Hopf algebra coactions have been highlighted.
+One of the most general constructions of universal (co)acting bialgebras/Hopf algebras,
+performed in the setting of Ω-algebras, was introduced in [1] together with generalized
+duality results. Necessary and sufficient conditions for the existence of universal coacting
+bialgebras/Hopf algebras are provided, explaining in this general setting the need for
+assuming finite dimensionality in both Manin and Tambara’s papers.
+It is worth to point out that both Sweedler and Manin-Tambara’s constructions have
+a categorical interpretation. More precisely, for Tambara’s construction one considers
+the left adjoint, say a(A, −), of the tensor product endofunctor A ⊗ − on the category
+of k-algebras, where A is a finite dimensional associative algebra. Tambara’s universal
+coacting bialgebra is precisely a(A, A) which turns out to be naturally endowed with a
+bialgebra structure. Similarly, for an arbitrary associative algebra A, it can be proved
+that the contravariant functor Hom(−, A) taking coalgebras to (convolution) algebras
+has a right adjoint which hereafter we denote by M(A, −). As before, Sweedler’s uni-
+versal measuring bialgebra is exactly M(A, A) which again has a bialgebra structure.
+In this paper we deal with the representation theoretic version of Manin-Tambara’s con-
+struction in the Lie algebra setting. Our approach is a categorical one. More precisely,
+given two fixed Lie algebras g and h, with h finite dimensional, and the corresponding
+universal algebra A = A(h, g) (see[4]), we first show that the tensor product between
+an A-module U and a Lie h-module V can be endowed with a Lie g-module structure
+(Theorem 2.1). As a consequence, we are able to construct two ”tensor product” func-
+tors between the categories of Lie modules over h and g and respectively between the
+category of A-modules and the category of Lie g-modules. Under the appropriate finite
+dimensionality assumptions, the two functors mentioned above are proved to admit left
+adjoints. These left adjoints are given precisely by what we have called the universal
+Lie h-module and the universal A-module, respectively. The two universal modules are
+introduced in a constructive manner in Theorem 2.4 and Theorem 2.10. These are the
+counterparts for Lie and associative representations of Manin-Tambara’s constructions.
+Furthermore, the two aforementioned pairs of adjoint functors allow us to travel both
+ways between the representation categories of different algebraic structures, such as Lie
+and associative algebras, and to transfer certain properties which are usually preserved
+by left/right adjoints.
+1. Preliminaries
+This section will be used mostly as an opportunity to fix some notation and to provide
+certain useful references. Let us start with a few words on notation.
+1.1. Notational conventions. All vector spaces, (bi)linear maps, unadorned tensor
+products, Lie or associative algebras, bialgebras and so on are over an arbitrary com-
+mutative field k. All (co)associative (co)algebras are assumed to be (co)unital.
+The
+notation employed for coalgebras is standard: ∆ stands for the comultiplication and ε
+
+UNIVERSAL MODULES
+3
+for the counit. We use Sweedler’s notation with implied summation for both coalge-
+bras (resp. bialgebras), as in ∆(c) = c(1) ⊗ c2, and for comodule structures: a right
+C-comodule structure ρ on a vector space V will be denoted by ρ(v) = v(0) ⊗v(1). When
+we need to be precise, the structures involved will be adorned. δij denotes Kronecker’s
+symbol while IdX stands for the identity map on the set X.
+In the sequel, k[Xsi |s = 1, · · · , n, i ∈ I] denotes the usual polynomial algebra on vari-
+ables Xsi. We shall denote by Liek and ComAlgk the categories of Lie and commutative
+associative algebras, respectively. Given an associative algebra A and a Lie algebra g
+we denote by AM and gLM the categories of left A-modules and left Lie g-modules, re-
+spectively. Recall that a (left) Lie g-module is a vector space V equipped with a bilinear
+map ⇀: g × V → V such that for all x, y ∈ g and v ∈ V we have:
+[x, y] ⇀ v = x ⇀ (y ⇀ v) − y ⇀ (x ⇀ v).
+Throughout the paper, g and h will denote two arbitrary Lie algebras with h finite
+dimensional. Let {fi | i ∈ I} and {e1, · · · , en} be two fixed basis in g and h, respectively.
+We consider {τ s
+i,j | i, j, s = 1, · · · , n} to be the structure constants of h, i.e. for any i,
+j = 1, · · · , n we have:
+[ei, ej]h =
+n
+�
+s=1
+τ s
+i,j es.
+(1)
+Similarly, for any i, j ∈ I, let Bi,j ⊆ I be a finite subset of I such that for any i, j ∈ I
+we have:
+[fi, fj]g =
+�
+u∈Bi,j
+βu
+i,j fu.
+(2)
+1.2. The universal algebra of h and g. We recall briefly, for further use, the con-
+struction of the universal commutative algebra A(h, g) of two given Lie algebras h and
+g (recall that h is always assumed to be finite dimensional). It was first introduced in
+[4] in the more general setting of Leibniz algebras as the counterpart of Tambara’s con-
+struction ([16]). We restrict here to the Lie algebra version of the construction which
+can be summarized as follows. We have:
+A(h, g) := k[Xsi |s = 1, · · · , n, i ∈ I]/J
+(3)
+where J is the ideal generated by all polynomials of the form
+P (h, g)
+(a,i,j) :=
+�
+u∈Bi,j
+βu
+i,j Xau −
+n
+�
+s,t=1
+τ a
+s,t XsiXtj,
+for all a = 1, · · · , n and i, j ∈ I.
+(4)
+When working in the universal algebra A(h, g), we denote by xsi := �
+Xsi the class of Xsi.
+Consequently, the following relations hold in A(h, g):
+�
+u∈Bi,j
+βu
+i,j xau =
+n
+�
+s,t=1
+τ a
+s,t xsixtj,
+for all a = 1, · · · , n, and i, j ∈ I.
+(5)
+When the (finite dimensional) Lie algebra h is fixed, the universal algebra construction
+gives rise to a functor A(h, −): Liek → ComAlgk which turns out to be the left adjoint
+
+4
+A. L. AGORE
+of the tensor product h ⊗ −: ComAlgk → Liek (see [4, Theorem 2.1]), where for any
+commutative algebra X the tensor product h ⊗ X is endowed with the current Lie
+algebra structure. In order to avoid dealing with cumbersome notation, when there is no
+fear of confusion, we denote A = A(h, g). Furthermore, If h = g, then the corresponding
+universal algebra A(h, h) will be denoted simply by B. The notation is meant to highlight
+the fact that B is a bialgebra; in fact, it admits a unique bialgebra structure such that h
+becomes a right B-comodule with respect to ηh : h → h⊗B where η: 1Liek → h⊗A(h, −)
+denotes the unit of the adjunction between A(h, −) and h ⊗ −.
+More precisely, the
+comultiplication and the counit on B are given for any i, j = 1, · · · , n by
+∆(xij) =
+n
+�
+s=1
+xis ⊗ xsj
+and
+ε(xij) = δi,j1k
+(6)
+For basic categorical concepts we refer the reader to [10] and for unexplained notions
+pertaining to Lie and Hopf algebras to [9] and [15], respectively.
+2. Universal modules
+Our first important result provides a way of defining a Lie g-module structure on the
+tensor product between a Lie h-module and an A-module.
+Theorem 2.1. Let (U, ↷) ∈ hLM be a Lie h-module and (V, ·) ∈ AM an A-module.
+Then (U ⊗ V, ⇀) ∈ gLM is a Lie g-module where the action of g on U ⊗ V is given for
+all i ∈ I, l ∈ U and t ∈ V by:
+fi ⇀ (l ⊗ t) =
+n
+�
+j=1
+(ej ↷ l) ⊗ (xji · t)
+(7)
+Proof. Indeed, having in mind that (U, ↷) is a Lie module and A = A(h, g) is a com-
+mutative algebra, we have:
+[fi, fj] ⇀ (l ⊗ t)
+(2)
+=
+�
+u∈Bi,j
+βu
+i,j fu ⇀ (l ⊗ t)
+(7)
+=
+�
+u∈Vi,j,r=1,n
+βu
+i,j (er ↷ l) ⊗ (xru · t)
+=
+�
+r=1,n
+(er ↷ l) ⊗
+� �
+u∈Bi,j
+βu
+i,j xru
+�
+·t
+(5)
+=
+�
+s,p,r=1,n
+τ r
+s,p (er ↷ l) ⊗ (xsixpj) · t
+=
+�
+s,p=1,n
+� n
+�
+r=1
+τ r
+s,p er
+�
+↷ l ⊗ (xsixpj) · t
+(1)
+=
+�
+s,p=1,n
+[es, ep] ↷ l ⊗ (xsixpj) · t
+=
+�
+s,p=1,n
+es ↷ (ep ↷ l) ⊗ xsi · (xpj · t) −
+�
+s,p=1,n
+ep ↷ (es ↷ l) ⊗ xpj · (xsi · t)
+(7)
+= fi ⇀
+n
+�
+p=1
+(ep ↷ l) ⊗ (xpj · t) − fj ⇀
+n
+�
+s=1
+(es ↷ l) ⊗ (xsi · t)
+(7)
+= fi ⇀
+�
+fj ⇀ (l ⊗ t)
+�
+− fj ⇀
+�
+fi ⇀ (l ⊗ t)
+�
+
+UNIVERSAL MODULES
+5
+for all i, j ∈ I and l ∈ U, t ∈ V , i.e. (U ⊗ V, ⇀) is a left Lie g-module.
+□
+Inspired by Theorem 2.1 we can consider two types of universal modules.
+2.1. The universal A-module. The first such universal module is associated with a
+Lie h-module and a Lie g-module as follows:
+Definition 2.2. Given a Lie h-module U and a Lie g-module Z, the universal A-module
+of U and Z is a pair
+�
+U(U, Z), ρU(U, Z)
+�
+consisting of an A-module U(U, Z) and a mor-
+phism of Lie g-modules ρU(U, Z) : Z → U ⊗ U(U, Z) such that for any other pair (X, f)
+consisting of an A-module X and a morphism of Lie g-modules f : Z → U ⊗X, there ex-
+ists a unique morphism of A-modules g: U(U, Z) → X such that the following diagram
+is commutative:
+Z
+ρU(U, Z)
+�
+f
+�❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+U ⊗ U(U, Z)
+IdU⊗g
+�
+U ⊗ X
+(8)
+In other words, the above definition is saying that, when it exists, the universal A-module
+of U and Z is in fact the initial object of the category whose objects are pairs (X, f)
+consisting of an A-module X and a morphism of Lie g-modules f : Z → U ⊗ X, while
+morphisms between two such objects (X, f) and (X′, f ′) are defined to be A-module
+maps g: X → X′ satisfying (IdU ⊗ g) ◦ f = f ′.
+As direct consequences of the above definition, we obtain the following:
+Corollary 2.3. Let U be a Lie h-module. Then, for all Lie g-modules Z and all A-
+modules X, we have a bijective correspondence between:
+(1) Lie g-module maps f : Z → U ⊗ X;
+(2) A-module maps g: U(U, Z) → X.
+Under the appropiate finite-dimensionality assumptions required for all Manin-Tambara
+type constructions, the universal A-module introduced in Definition 2.2 exists:
+Theorem 2.4. If U is a finite dimensional Lie h-module then the universal A -module
+of U and any other Lie g-module Z exists.
+Proof. Let {u1, · · · , um}, m ∈ N∗, be a k-basis of the Lie module U and denote by ωt
+ij ∈ k
+the structure constants of U with respect to its Lie h-module structure ↷, i.e. for all
+i = 1, · · · , n, j = 1, · · · , m we have:
+ei ↷ uj =
+m
+�
+s=1
+ωs
+i,j us
+(9)
+Furthermore, consider {zr | r ∈ J} to be a k-basis for the arbitrary Lie g-module Z and
+if ↬ denotes its Lie module structure, then for all j ∈ I and r ∈ J we can find a finite
+
+6
+A. L. AGORE
+subset Tj,r of J such that:
+fj ↬ zr =
+�
+l∈Tj,r
+ηl
+j,r zl
+(10)
+where ηl
+j,r ∈ k for all j ∈ I, r ∈ J, and l ∈ Tj,r.
+Consider now T (U, Z) to be the free A-module on the set {Yij | i = 1, · · · , m, j ∈ J}
+and denote by U(U, Z) the quotient of T (U, Z) by its A-submodule generated by the
+following elements:
+�
+p∈Tj,i
+ηp
+j,i Ysp −
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj • Yti
+(11)
+for all s = 1, · · · , m, i ∈ J and j ∈ I, where • denotes the A-module action on T (U, Z).
+Denoting ytj := �
+Ytj, where �
+Ytj stands for the equivalence class of Ytj in the quotient
+module U(U, Z), it follows that the relations below hold in U(U, Z):
+�
+p∈Tj,i
+ηp
+j,i ysp =
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj • yti
+(12)
+for all s = 1, · · · , m, i ∈ J and j ∈ I.
+Furthermore, we can define a morphism of Lie g-modules ρU(U, Z): Z → U ⊗ U(U, Z) as
+follows:
+ρU(U, Z)(zr) :=
+m
+�
+s=1
+us ⊗ ysr,
+for all r ∈ J.
+(13)
+It follows now that for all j ∈ I and i ∈ J we have:
+ρU(U, Z)(fj ↬ zi)
+(10)
+= ρU(U,Z)
+� �
+p∈Tj,i
+ηp
+ji zp
+�
+=
+�
+p∈Tj,i
+m
+�
+s=1
+ηp
+ji us ⊗ ysp =
+m
+�
+s=1
+�
+us ⊗
+�
+p∈Tj,i
+ηp
+ji ysp
+�
+(12)
+=
+m
+�
+s,t=1
+n
+�
+r=1
+ωs
+r,t us ⊗ xrj • yti =
+m
+�
+t=1
+n
+�
+r=1
+� m
+�
+s=1
+ωs
+r,t us
+�
+⊗ xrj • yti
+(9)
+=
+m
+�
+t=1
+n
+�
+r=1
+er ↷ ut ⊗ xrj • yti
+(7)
+=
+m
+�
+t=1
+fj ⇀ (ut ⊗ yti) = fj ⇀
+m
+�
+t=1
+ut ⊗ yti
+(13)
+= fj ⇀ ρU(U, Z)(zi)
+which shows that ρU(U, Z) is indeed a Lie g-modules map.
+We will show that the pair
+�
+U(U, Z), ρU(U, Z)
+�
+constructed above is in fact the universal
+A-module of U and Z. To this end, consider a pair (X, f) consisting of an A-module X
+and a morphism of Lie g-modules f : Z → U ⊗ X. Let {wsr | s = 1, · · · , m, r ∈ J} be a
+family of elements of X such that for all r ∈ J we have:
+g(zr) =
+m
+�
+s=1
+us ⊗ wsr
+(14)
+
+UNIVERSAL MODULES
+7
+Furthermore, as g: Z → U ⊗ X is a Lie g-modules map, a straightforward computation
+shows that the following compatibilities hold for all s = 1, · · · , m, i ∈ J and j ∈ I:
+�
+p∈Tj,i
+ηp
+j,i wsp =
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj · wti
+(15)
+where · denotes the A-module action on X.
+The universal property of the free module yields a unique A-module map g: T (U, Z) → X
+such that g(Ysr) = wsr, for all s = 1, · · · , m and r ∈ J. Moreover, Ker(g) contains the A-
+submodule of T (U, Z) generated by the elements listed in (11). Indeed, as g : U(U, Z) →
+X is a morphism of A-modules we have:
+g
+� �
+p∈Tj,i
+ηp
+j,i Ysp −
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj • Yti
+�
+=
+�
+p∈Tj,i
+ηp
+j,i wsp −
+m
+�
+t=1
+n
+�
+r=1
+ωs
+r,t xrj · wti
+(15)
+= 0
+for all s = 1, · · · , m, i ∈ J and j ∈ I. This shows that there exists a unique A-modules
+map g: U(U, Z) → X such that g(ysr) = zsr, for all s = 1, · · · , m and r ∈ J. This
+implies that for all r ∈ J we have:
+�
+IdU ⊗ g
+�
+◦ ρU(U, Z)(zr) =
+�
+IdU ⊗ g
+�� m
+�
+s=1
+us ⊗ ysr
+�
+=
+m
+�
+s=1
+us ⊗ wsr
+(14)
+= g(zr)
+which means precisely that diagram (8) is commutative. Moreover, g is obviously the
+unique A-modules map with this property and the proof is now finished.
+□
+The case g = h. Particularizing the results of Section 2 for g = h, where h is the finite
+dimensional Lie algebra defined in (1), leads to the following interesting consequences.
+According to the discussion in Preliminaries, the universal algebra A(h, h) denoted by B
+is in this case a bialgebra with coalgebra structure depicted in (6). This allows us to see
+the tensor product U(U, Z) ⊗ U(U, Z) as well as the base field k as B-modules via the
+comultiplication and the counit of B as follows:
+xij ∗ (y ⊗ t) =
+n
+�
+t=1
+xit • y ⊗ xtj • t
+(16)
+xij · α = δijα
+(17)
+for all xij ∈ B, y, t ∈ U(U, Z) and α ∈ k, where • denotes the B-module strucuture on
+U(U, Z) as in the proof of Theorem 2.4.
+First we show that if U is a finite dimensional Lie h-module as considered in (9), then
+the B-module U(U, U) denoted by U(U) admits a coalgebra structure with respect to
+which
+�
+U, ρU(U)
+�
+becomes a right U(U)-comodule.
+Proposition 2.5. Let U be a finite dimensional Lie h-module. There exists a unique
+coalgebra structure on U(U) such that
+�
+U, ρU(U)
+�
+becomes a right U(U)-comodule.
+Proof. In particular both U(U) ⊗ U(U) and k are B-modules via the formulas (16) and
+(17) respectively. Therefore, U ⊗ U(U) ⊗ U(U) and U ⊗ k are Lie h-modules via (7).
+Furthermore, it can be easily checked that the maps
+�
+ρU(U) ⊗ IdU(U)
+�
+◦ρU(U) : U → U ⊗
+
+8
+A. L. AGORE
+U(U) ⊗ U(U) and canU : U → U ⊗ k are morphisms of Lie h-modules, where canU : U →
+U ⊗ k is the canonical isomorphism. Now Definition 2.2 yields a unique B-modules map
+∆: U(U) → U(U) ⊗ U(U) such that the following diagram is commutative:
+U
+ρU(U)
+�
+�
+ρU(U)⊗IdU(U)
+�
+◦ρU(U)
+�❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+❆
+U ⊗ U(U)
+IdU ⊗∆
+�
+U ⊗ U(U) ⊗ U(U)
+Similarly, we obtain a unique B-modules map ε: U(U) → k such that the following
+diagram is commutative:
+U
+ρU(U) �
+canU
+�■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+U ⊗ U(U)
+IdU⊗ε
+�
+U ⊗ k
+A straightforward computation shows that the commutativity of the two diagrams above
+imply that ∆ and ε take the following form for all l, t = 1, · · · , m:
+∆(ylt) =
+m
+�
+s=1
+yls ⊗ yst,
+ε(ylt) = δlt1k.
+It is now obvious that
+�
+U(U), ∆, ε
+�
+form a coalgebra. Finally, by the commutativity of
+the two diagrams above we obtain that
+�
+U, ρU(U)
+�
+is a right U(U)-comodule.
+□
+Remark 2.6. It is worth pointing out that with the coalgebra structure introduced
+above, U(U) becomes a B-module coalgebra. Indeed, having in mind that both ∆ and ε
+are B-module maps, we have:
+∆(xab • ylt) = xab ∗ ∆(ylt) = xab ∗
+� m
+�
+s=1
+yls ⊗ yst
+�(16)
+=
+n
+�
+c=1
+m
+�
+s=1
+xac • yls ⊗ xcb • yst
+= (xab)(1) • (ylt)(1) ⊗ (xab)(2) • (ylt)(2)
+and
+ε(xab • ylt) = xab · ε(ylt)
+(17)
+= δab ε(ylt) = ε(xab) ε(ylt).
+This shows that • is a coalgebra map, as desired.
+It turns out that the pair
+�
+U(U), ρU(U)
+�
+is universal in the following way:
+Proposition 2.7. For any coalgebra X with a B-module structure and any Lie h-module
+morphism ψ: U → U ⊗X which makes U into a right X-comodule, there exists a unique
+
+UNIVERSAL MODULES
+9
+B-modules and coalgebra morphism θ: U(U) → X such that the following diagram is
+commutative:
+U
+ρU(U) �
+ψ
+�■
+■
+■
+■
+■
+■
+■
+■
+■
+■
+U ⊗ U(U)
+IdU ⊗θ
+�
+U ⊗ X
+Proof. In light of Definition 2.2, such a unique A-modules map θ exists. We are left to
+show that θ is also a coalgebra map. From the proof of Theorem 2.4 we know that θ is
+defined for all l, t = 1, · · · , m by θ(ylt) = zlt where zlt are elements of X such that for all
+r = 1, · · · , m we have ψ(ur) = �m
+s=1 us ⊗ zsr. As (U, ψ) is a right comodule, we obtain:
+∆(zlt) =
+m
+�
+s=1
+zls ⊗ zst,
+ε(zlt) = δlt1k.
+To this end, we have:
+∆
+�
+θ(ylt)
+�
+= ∆(zlt) =
+m
+�
+s=1
+zls ⊗ zst =
+m
+�
+s=1
+θ(yls) ⊗ θ(yst) = (θ ⊗ θ) ◦ ∆(ylt)
+Similarly one can check that ε◦θ = ε which shows that θ is indeed a coalgebra map.
+□
+2.2. The universal h-module. The second type of universal module one can consider
+is the following:
+Definition 2.8. Given an A-module V and a Lie g-module W, the universal Lie h-
+module of V and W is a pair
+�
+V(V, W), τV(V, W )
+�
+consisting of a Lie h-module V(V, W)
+and a morphism of Lie g-modules τV(V, W ): W → V(V, W)⊗V such that for any other pair
+(Y, f) consisting of a Lie h-module Y and a morphism of Lie g-modules f : W → Y ⊗ V ,
+there exists a unique morphism of Lie h-modules g: V(V, W) → Y such that the following
+diagram is commutative:
+W
+τV(V, W )
+�
+f
+�❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+❘
+V(V, W) ⊗ V
+g⊗IdV
+�
+Y ⊗ V
+(18)
+The universal Lie h-module of V and W, when it exists, can again be seen as the initial
+object of the category whose objects are pairs (Y, f) consisting of a Lie h-module Y
+and a morphism of Lie g-modules f : W → Y ⊗ V , while morphisms between two such
+objects (Y, f) and (Y ′, f ′) are defined to be Lie h-module maps g: Y → Y ′ satisfying
+(g ⊗ IdV ) ◦ f = f ′.
+Corollary 2.9. Let V be an A-module. Then, for all Lie g-modules W and all Lie
+h-modules Y , we have a bijective correspondence between:
+(1) Lie g-module maps f : W → Y ⊗ V ;
+(2) Lie h-module maps g: V(V, W) → Y .
+
+10
+A. L. AGORE
+The universal h-module introduced in Definition 2.8 also exists provided that the A-
+module V is finite dimensional.
+Theorem 2.10. If V is a finite dimensional A-module then the universal Lie h-module
+of V and any other Lie g-module W exists.
+Proof. As this proof is somewhat similar in spirit with the one of Theorem 2.4, we will be
+brief and provide only the main ingredients required for the construction of the universal
+Lie h-module.
+Let {v1, · · · , vl}, l ∈ N∗, be a k-basis of the finite dimensional A-module V and denote
+by γt
+r,i,j ∈ k the structure constants of V with respect to its A-module structure ·, i.e.
+for all r = 1, · · · , n, i ∈ I and j = 1, · · · , l we have:
+xri · vj =
+l
+�
+s=1
+γs
+r,i,j vs
+(19)
+Consider {wr | r ∈ T} to be a k-basis for W and if ⊲ denotes its Lie g-module structure,
+then for all j ∈ I and r ∈ T we can find a finite subset Sj,r of T such that:
+fj ⊲ wr =
+�
+p∈Sj,r
+σp
+j,r wp
+(20)
+where σp
+j,r ∈ k for all j ∈ I, r ∈ T, and p ∈ Sj,r.
+Now let S(V, W) be the free Lie h-module on the set {Yri | r ∈ T, i = 1, · · · , l} and
+denote by V(V, W) the quotient of S(V, W) by its Lie h-submodule generated by the
+following elements:
+�
+p∈Sj,r
+σp
+j,r Yps −
+l
+�
+k=1
+n
+�
+p=1
+γs
+p,j,k ep ◮ Yrk
+(21)
+for all s = 1, · · · , l, r ∈ T and j ∈ I, where ◮ denotes the h-module action on S(V, W).
+By denoting yri := �
+Yri, where �
+Yri stands for the equivalence class of Yri in the quotient
+module V(V, W), it follows that the relations below hold in V(V, W):
+�
+p∈Sj,r
+σp
+j,r yps =
+l
+�
+k=1
+n
+�
+t=1
+γs
+t,j,k et ◮ yrk
+(22)
+for all s = 1, · · · , l, r ∈ T and j ∈ I.
+It can now be easily seen, as in the proof of Theorem 2.4, that the pair (V(V, W), τV(V, W ))
+is the universal Lie h-module of V and W, where τV(V, W ): W → V(V, W) ⊗ V is the
+morphism of Lie g-modules defined for all r ∈ T as follows:
+τV(V, W )(wr) :=
+l
+�
+s=1
+yrs ⊗ vs.
+(23)
+□
+
+UNIVERSAL MODULES
+11
+3. Functors between module categories
+In this section we show that the two universal module constructions previously introduced
+are functorial and, moreover, if certain conditions are fulfilled the corresponding functors
+admit right adjoints. We start, however, by stating the following easy consequence of
+Theorem 2.1:
+Proposition 3.1. Let (U, ↷) ∈ hLM and (V, ·) ∈ AM. Then:
+1) We have a functor U ⊗ −: AM → gLM from the category of A-modules to the
+category of Lie g-modules;
+2) We have a functor − ⊗ V : hLM → gLM between the categories of Lie modules
+over h and g respectively.
+Proof. In light of Theorem 2.1, we are only left to show that morphisms behave well with
+respect to the corresponding associative or Lie module structures. We will treat only the
+first statement and leave the second one to the reader. To this end, consider (V, ·) and
+(V ′, •) two A-modules, ⇀ and ⇀′ the corresponding induced Lie g-module actions via
+(7) and g: V → V ′ a morphism in AM . Then, for all i ∈ I, l ∈ U and t ∈ V we have:
+(IdU ⊗ g)
+�
+fi ⇀ (l ⊗ t)
+�(7)
+=
+n
+�
+j=1
+(ej ↷ l) ⊗ g(xji · t) =
+n
+�
+j=1
+(ej ↷ l) ⊗ xji • g(t)
+(7)
+= fi ⇀′ �
+l ⊗ g(t)
+�
+□
+We consider now the universal module functors:
+Theorem 3.2. Let U be a finite dimensional Lie h-module and V a finite dimensional
+A-module.
+(1) There exists a functor UU : gLM → AM defined as follows for all Lie g-modules
+X, Y and all morphisms f : X → Y in gLM:
+UU(X) = U(U, X),
+UU(f) = f
+where f : U(U, X) → U(U, Y ) is the unique A-modules morphism which makes
+the following diagram commutative:
+X
+ρU(U, X)
+�
+ρU(U, Y )◦f
+�◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+U ⊗ U(U, X)
+IdU⊗f
+�
+U ⊗ U(U, Y )
+(24)
+(2) There exists a functor VV : gLM → hLM defined as follows for all Lie g-modules
+X, Y and all morphisms f : X → Y in gLM:
+VV (X) = V(V, X),
+VV (f) = f
+
+12
+A. L. AGORE
+where f : V(V, X) → V(V, Y ) is the unique morphism of Lie h-modules which
+makes the following diagram commutative:
+X
+τV(V, X)
+�
+τV(V, Y )◦f
+�◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+V(V, X) ⊗ V
+f⊗IdV
+�
+V(V, Y ) ⊗ V
+(25)
+Proof. As the result follows in a straightforward manner by a standard category the-
+ory argument, we only sketch the proof of the first assertion. Indeed, if f = IdX then
+IdU(U, X) is obviously the unique A-modules morphism which makes diagram (24) com-
+mute and therefore UU(IdX) = IdU(U, X). Moreover, if f : X → Y and g: Y → W are two
+morphisms in gLM, then g ◦f : U(U, X) → U(U, W) is obviously the unique A-modules
+morphism which makes the following diagram commutative:
+Z
+ρU(U, X)
+�
+ρU(U, W )◦g◦f
+�◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+◗
+U ⊗ U(U, X)
+IdU⊗
+�
+g◦f
+�
+�
+U ⊗ U(U, W)
+and we can conclude that UU(g ◦ f) = UU(g) ◦ UU(f), as desired.
+□
+Under the appropriate finite-dimensionality assumptions, the functors constructed in
+Proposition 3.1 are right adjoints to the universal module functors:
+Theorem 3.3. Let (U, ↷) be a finite dimensional Lie h-module and (V, ·) a finite di-
+mensional A-module. Then:
+1) The following functors form an adjunction:
+UU : gLM → AM,
+U ⊗ −: AM → gLM;
+2) Similarly, the following functors also form an adjunction:
+VV : gLM → hLM,
+− ⊗ V : hLM → gLM.
+Proof. 1) As pointed out in Corollary 2.3, for all Lie g-modules Z and all A-modules
+X, there is a bijection between HomAM
+�
+UU(Z), X
+�
+and HomgLM (Z, U ⊗ X) given as
+follows for all morphisms of A-modules θ: UU(Z) → X:
+ΓZ,X : HomAM (UU(Z), X) → HomgLM (Z, U ⊗ X),
+ΓZ,X(θ) = (IdU ⊗ θ) ◦ ρU(U, Z).
+The desired conclusion now follows by showing that the above bijection is natural in
+both variables. This can be easily proved by a straightforward diagram chase and is left
+to the reader.
+2) Using now Corollary 2.9, for all Lie g-modules W and all Lie h-modules Z, we obtain
+a bijection between HomhLM
+�
+VV (W), Z
+�
+and HomgLM (W, Z ⊗ V ) defined as follows
+
+UNIVERSAL MODULES
+13
+for all morphisms of Lie h-modules θ: VV (W) → Z:
+ΓW,Z : HomhLM
+�
+VV (W), Z
+�
+→ HomgLM (W, Z ⊗ V ),
+ΓW,Z(θ) = (θ ⊗ IdV ) ◦ ρV(V, W ).
+□
+In particular, the two pairs of adjoint functors allow us to travel both ways between the
+representation categories of the two (arbitrary) Lie algebras h and g and respectively
+between the representation category of the associative algebra A and the representation
+category of the Lie algebra g.
+Example 3.4. Let ρi : g ⊗ Wi → Wi be representations of g, where i = 1, 2. By the
+colimit preservation property of left adjoints we can easily conclude that for any finite
+dimensional Lie h-module U, UU(W1) ⊕ UU(W2) is the direct sum of the A-modules
+UU(W1) and UU(W2).
+Similarly, for any finite dimensional A-module V , VV (W1) ⊕
+UU(W2) is the direct sum of the Lie h-modules UU(W1) and UU(W2). This can be easily
+extended to an arbitrary family of representations.
+References
+[1] Agore, A.L., Gordienko, A.S., Vercruysse, J. - V -universal Hopf algebras (co)acting on Ω-algebras,
+Commun. Contemp. Math. 25 (2023), 2150095.
+[2] Agore, A.L. - Universal coacting Poisson Hopf algebras, Manuscripta Math. 165 (2021), 255–268.
+[3] Agore, A.L., Gordienko, A.S., Vercruysse, J. - Equivalences of (co)module algebra structures over
+Hopf algebras, J. Noncommut. Geom., 15 (2021), 951–993.
+[4] Agore, A.L., Militaru, G. - A new invariant for finite dimensional Leibniz/Lie algebras, J. Algebra
+562 (2020), 390–409.
+[5] Ardizzoni, A., El Kaoutit, L., Menini, C. - Categories of comodules and chain complexes of modules,
+Internat. J. Math. 23 (2012), 1250109
+[6] Bhattacharjee, S., Chirvˇasitu, A., Goswami, D. - Quantum Galois groups of subfactors, Internat. J.
+Math. 33 (2022), 2250013.
+[7] Chirvˇasitu, A., Walton, C., Wang, X. - On quantum groups associated to a pair of preregular forms,
+J. Noncommut. Geom. bf 13 (2019), 115—159.
+[8] Hyland, M., Lopez Franco, I., Vasilakopoulou, C. - Hopf measuring comonoids and enrichment,
+Proc. Lond. Math. Soc. 115 (2017), 1118—1148.
+[9] Jacobson, N. – Lie algebras, Dover Publications, NY, 1962.
+[10] Mac Lane, S. - Categories for the Working Mathematician, Graduate Texts in Mathematics 5,
+Springer, 1998.
+[11] Manin, Yu. I. - Quantum groups and noncommutative geometry, Universite de Montreal, Centre de
+Recherches Mathematiques, Montreal, QC, 1988.
+[12] Militaru, G. - The automorphisms group and the classification of gradings of finite dimensional
+associative algebras, Results Math. 77 (2022).
+[13] Raedschelders, T., Van den Bergh, M. - The Manin Hopf algebra of a Koszul Artin-Schelter regular
+algebra is quasi-hereditary, Adv. Math. 305 (2017), 601-–660.
+[14] Rodrıiguez-Romo, S., Taft, E. - Some quantum-like Hopf algebras which remain noncommutative
+when q = 1, Lett. Math. Phys. 61(2002), 41-–50.
+[15] Sweedler, M.E. - Hopf Algebras, Benjamin New York, 1969.
+[16] Tambara, D. - The coendomorphism bialgebra of an algebra. J. Fac. Sci. Univ. Tokyo Math. 37
+(1990), 425–456.
+
+14
+A. L. AGORE
+Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium
+Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700
+Bucharest, Romania
+Email address: ana.agore@gmail.com
+

0NE1T4oBgHgl3EQfRQO7/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf,len=433
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='03051v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='RA]  8 Jan 2023  FUNCTORS BETWEEN REPRESENTATION CATEGORIES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  UNIVERSAL MODULES  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' AGORE  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let g and h be two Lie algebras with h finite dimensional and consider  A = A(h, g) to be the corresponding universal algebra as introduced in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Given an  A-module U and a Lie h-module V we show that U ⊗ V can be naturally endowed  with a Lie g-module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' This gives rise to a functor between the category of Lie  h-modules and the category of Lie g-modules and, respectively, to a functor between  the category of A-modules and the category of Lie g-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Under some finite  dimensionality assumptions, we prove that the two functors admit left adjoints which  leads to the construction of universal A-modules and universal Lie h-modules as the  representation theoretic counterparts of Manin-Tambara’s universal coacting objects  [11, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Introduction  The universal coacting bialgebra/Hopf algebra on a finite dimensional (graded) asso-  ciative algebra originates in the work of Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Manin ([11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The importance of this  construction became obvious mostly due to its interaction with non-commutative geom-  etry where it is seen as some sort of symmetry group (see [13] for more details on this  view point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The non-graded version of this construction appeared a few years later in a  paper by D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Tambara ([16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' However, as remarked in [16], the universal coacting bialge-  bra is in fact the dual of the so-called universal measuring bialgebra introduced by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Sweedler in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' We should note that, unlike Manin-Tambara’s construction, Sweedler’s  universal measuring bialgebra/Hopf algebra exists even in the infinite-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  In recent years, universal (co)acting objects have been considered in various settings  and for different purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' For instance, [8] extends Sweedler’s construction to monoids  in a braided monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' On the other hand, the Manin-Tambara construction  was introduced in the setting of Poisson algebras ([2]), finite index-subfactors ([6]), su-  perpotential algebras ([7]), polynomial algebras ([14]), bialgebroids ([5]) or Lie/Leibniz  algebras ([4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The corresponding universal coacting bialgebras/Hopf algebras, which in  certain cases carry some extra structure (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' a Poisson Hopf algebra structure as in [2]),  seem to play a prominent role in solving other seemingly unrelated problems such as  the classification of gradings on various kinds of algebras ([4, 12]), the description of the  automorphisms group of certain algebraic structures ([4]) and even in quantum Galois  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' 16D90, 16T05, 17A32, 17B10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' universal module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  This work was supported by a grant of Romanian Ministry of Research, Innovation and Digitization,  CNCS/CCCDI – UEFISCDI, project number PN-III-P4-ID-PCE-2020-0458, within PNCDI III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  1  2  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' AGORE  theory ([6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Another related universal (co)acting construction was considered in [3] as  the Hopf algebraic analogue of the universal group of a grading and its connections to  the problem of classifying Hopf algebra coactions have been highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  One of the most general constructions of universal (co)acting bialgebras/Hopf algebras,  performed in the setting of Ω-algebras, was introduced in [1] together with generalized  duality results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Necessary and sufficient conditions for the existence of universal coacting  bialgebras/Hopf algebras are provided, explaining in this general setting the need for  assuming finite dimensionality in both Manin and Tambara’s papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  It is worth to point out that both Sweedler and Manin-Tambara’s constructions have  a categorical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' More precisely, for Tambara’s construction one considers  the left adjoint, say a(A, −), of the tensor product endofunctor A ⊗ − on the category  of k-algebras, where A is a finite dimensional associative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Tambara’s universal  coacting bialgebra is precisely a(A, A) which turns out to be naturally endowed with a  bialgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Similarly, for an arbitrary associative algebra A, it can be proved  that the contravariant functor Hom(−, A) taking coalgebras to (convolution) algebras  has a right adjoint which hereafter we denote by M(A, −).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' As before, Sweedler’s uni-  versal measuring bialgebra is exactly M(A, A) which again has a bialgebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  In this paper we deal with the representation theoretic version of Manin-Tambara’s con-  struction in the Lie algebra setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Our approach is a categorical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' More precisely,  given two fixed Lie algebras g and h, with h finite dimensional, and the corresponding  universal algebra A = A(h, g) (see[4]), we first show that the tensor product between  an A-module U and a Lie h-module V can be endowed with a Lie g-module structure  (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' As a consequence, we are able to construct two ”tensor product” func-  tors between the categories of Lie modules over h and g and respectively between the  category of A-modules and the category of Lie g-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Under the appropriate finite  dimensionality assumptions, the two functors mentioned above are proved to admit left  adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' These left adjoints are given precisely by what we have called the universal  Lie h-module and the universal A-module, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The two universal modules are  introduced in a constructive manner in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='4 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' These are the  counterparts for Lie and associative representations of Manin-Tambara’s constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Furthermore, the two aforementioned pairs of adjoint functors allow us to travel both  ways between the representation categories of different algebraic structures, such as Lie  and associative algebras, and to transfer certain properties which are usually preserved  by left/right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Preliminaries  This section will be used mostly as an opportunity to fix some notation and to provide  certain useful references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let us start with a few words on notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Notational conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' All vector spaces, (bi)linear maps, unadorned tensor  products, Lie or associative algebras, bialgebras and so on are over an arbitrary com-  mutative field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' All (co)associative (co)algebras are assumed to be (co)unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  The  notation employed for coalgebras is standard: ∆ stands for the comultiplication and ε  UNIVERSAL MODULES  3  for the counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' We use Sweedler’s notation with implied summation for both coalge-  bras (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' bialgebras), as in ∆(c) = c(1) ⊗ c2, and for comodule structures: a right  C-comodule structure ρ on a vector space V will be denoted by ρ(v) = v(0) ⊗v(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' When  we need to be precise, the structures involved will be adorned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' δij denotes Kronecker’s  symbol while IdX stands for the identity map on the set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  In the sequel, k[Xsi |s = 1, · · · , n, i ∈ I] denotes the usual polynomial algebra on vari-  ables Xsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' We shall denote by Liek and ComAlgk the categories of Lie and commutative  associative algebras, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Given an associative algebra A and a Lie algebra g  we denote by AM and gLM the categories of left A-modules and left Lie g-modules, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Recall that a (left) Lie g-module is a vector space V equipped with a bilinear  map ⇀: g × V → V such that for all x, y ∈ g and v ∈ V we have:  [x, y] ⇀ v = x ⇀ (y ⇀ v) − y ⇀ (x ⇀ v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Throughout the paper, g and h will denote two arbitrary Lie algebras with h finite  dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let {fi | i ∈ I} and {e1, · · · , en} be two fixed basis in g and h, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  We consider {τ s  i,j | i, j, s = 1, · · · , n} to be the structure constants of h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' for any i,  j = 1, · · · , n we have:  [ei, ej]h =  n  �  s=1  τ s  i,j es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  (1)  Similarly, for any i, j ∈ I, let Bi,j ⊆ I be a finite subset of I such that for any i, j ∈ I  we have:  [fi, fj]g =  �  u∈Bi,j  βu  i,j fu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  (2)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The universal algebra of h and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' We recall briefly, for further use, the con-  struction of the universal commutative algebra A(h, g) of two given Lie algebras h and  g (recall that h is always assumed to be finite dimensional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' It was first introduced in  [4] in the more general setting of Leibniz algebras as the counterpart of Tambara’s con-  struction ([16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' We restrict here to the Lie algebra version of the construction which  can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' We have:  A(h, g) := k[Xsi |s = 1, · · · , n, i ∈ I]/J  (3)  where J is the ideal generated by all polynomials of the form  P (h, g)  (a,i,j) :=  �  u∈Bi,j  βu  i,j Xau −  n  �  s,t=1  τ a  s,t XsiXtj,  for all a = 1, · · · , n and i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  (4)  When working in the universal algebra A(h, g), we denote by xsi := �  Xsi the class of Xsi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Consequently, the following relations hold in A(h, g):  �  u∈Bi,j  βu  i,j xau =  n  �  s,t=1  τ a  s,t xsixtj,  for all a = 1, · · · , n, and i, j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  (5)  When the (���nite dimensional) Lie algebra h is fixed, the universal algebra construction  gives rise to a functor A(h, −): Liek → ComAlgk which turns out to be the left adjoint  4  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' AGORE  of the tensor product h ⊗ −: ComAlgk → Liek (see [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1]), where for any  commutative algebra X the tensor product h ⊗ X is endowed with the current Lie  algebra structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' In order to avoid dealing with cumbersome notation, when there is no  fear of confusion, we denote A = A(h, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Furthermore, If h = g, then the corresponding  universal algebra A(h, h) will be denoted simply by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The notation is meant to highlight  the fact that B is a bialgebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' in fact, it admits a unique bialgebra structure such that h  becomes a right B-comodule with respect to ηh : h → h⊗B where η: 1Liek → h⊗A(h, −)  denotes the unit of the adjunction between A(h, −) and h ⊗ −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  More precisely, the  comultiplication and the counit on B are given for any i, j = 1, · · · , n by  ∆(xij) =  n  �  s=1  xis ⊗ xsj  and  ε(xij) = δi,j1k  (6)  For basic categorical concepts we refer the reader to [10] and for unexplained notions  pertaining to Lie and Hopf algebras to [9] and [15], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Universal modules  Our first important result provides a way of defining a Lie g-module structure on the  tensor product between a Lie h-module and an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let (U, ↷) ∈ hLM be a Lie h-module and (V, ·) ∈ AM an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Then (U ⊗ V, ⇀) ∈ gLM is a Lie g-module where the action of g on U ⊗ V is given for  all i ∈ I, l ∈ U and t ∈ V by:  fi ⇀ (l ⊗ t) =  n  �  j=1  (ej ↷ l) ⊗ (xji · t)  (7)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' having in mind that (U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' ↷) is a Lie module and A = A(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' g) is a com-  mutative algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' we have:  [fi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' fj] ⇀ (l ⊗ t)  (2)  =  �  u∈Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='j  βu  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='j fu ⇀ (l ⊗ t)  (7)  =  �  u∈Vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='r=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='n  βu  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='j (er ↷ l) ⊗ (xru · t)  =  �  r=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='n  (er ↷ l) ⊗  � �  u∈Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='j  βu  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='j xru  �  t  (5)  =  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='r=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='n  τ r  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='p (er ↷ l) ⊗ (xsixpj) · t  =  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='n  � n  �  r=1  τ r  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='p er  �  ↷ l ⊗ (xsixpj) · t  (1)  =  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='n  [es,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' ep] ↷ l ⊗ (xsixpj) · t  =  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='n  es ↷ (ep ↷ l) ⊗ xsi · (xpj · t) −  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='p=1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='n  ep ↷ (es ↷ l) ⊗ xpj · (xsi · t)  (7)  = fi ⇀  n  �  p=1  (ep ↷ l) ⊗ (xpj · t) − fj ⇀  n  �  s=1  (es ↷ l) ⊗ (xsi · t)  (7)  = fi ⇀  �  fj ⇀ (l ⊗ t)  �  − fj ⇀  �  fi ⇀ (l ⊗ t)  �  UNIVERSAL MODULES  5  for all i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' j ∈ I and l ∈ U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' t ∈ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' (U ⊗ V, ⇀) is a left Lie g-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  □  Inspired by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1 we can consider two types of universal modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The universal A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The first such universal module is associated with a  Lie h-module and a Lie g-module as follows:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Given a Lie h-module U and a Lie g-module Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' the universal A-module  of U and Z is a pair  �  U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z)  �  consisting of an A-module U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z) and a mor-  phism of Lie g-modules ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z) : Z → U ⊗ U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z) such that for any other pair (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' f)  consisting of an A-module X and a morphism of Lie g-modules f : Z → U ⊗X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' there ex-  ists a unique morphism of A-modules g: U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z) → X such that the following diagram  is commutative:  Z  ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z)  �  f  �❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  U ⊗ U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z)  IdU⊗g  �  U ⊗ X  (8)  In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' the above definition is saying that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' when it exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' the universal A-module  of U and Z is in fact the initial object of the category whose objects are pairs (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' f)  consisting of an A-module X and a morphism of Lie g-modules f : Z → U ⊗ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' while  morphisms between two such objects (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' f) and (X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' f ′) are defined to be A-module  maps g: X → X′ satisfying (IdU ⊗ g) ◦ f = f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  As direct consequences of the above definition, we obtain the following:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let U be a Lie h-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Then, for all Lie g-modules Z and all A-  modules X, we have a bijective correspondence between:  (1) Lie g-module maps f : Z → U ⊗ X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  (2) A-module maps g: U(U, Z) → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Under the appropiate finite-dimensionality assumptions required for all Manin-Tambara  type constructions, the universal A-module introduced in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='2 exists:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' If U is a finite dimensional Lie h-module then the universal A -module  of U and any other Lie g-module Z exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let {u1, · · · , um}, m ∈ N∗, be a k-basis of the Lie module U and denote by ωt  ij ∈ k  the structure constants of U with respect to its Lie h-module structure ↷, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' for all  i = 1, · · · , n, j = 1, · · · , m we have:  ei ↷ uj =  m  �  s=1  ωs  i,j us  (9)  Furthermore, consider {zr | r ∈ J} to be a k-basis for the arbitrary Lie g-module Z and  if ↬ denotes its Lie module structure, then for all j ∈ I and r ∈ J we can find a finite  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' AGORE  subset Tj,r of J such that:  fj ↬ zr =  �  l∈Tj,r  ηl  j,r zl  (10)  where ηl  j,r ∈ k for all j ∈ I, r ∈ J, and l ∈ Tj,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Consider now T (U, Z) to be the free A-module on the set {Yij | i = 1, · · · , m, j ∈ J}  and denote by U(U, Z) the quotient of T (U, Z) by its A-submodule generated by the  following elements:  �  p∈Tj,i  ηp  j,i Ysp −  m  �  t=1  n  �  r=1  ωs  r,t xrj • Yti  (11)  for all s = 1, · · · , m, i ∈ J and j ∈ I, where • denotes the A-module action on T (U, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Denoting ytj := �  Ytj, where �  Ytj stands for the equivalence class of Ytj in the quotient  module U(U, Z), it follows that the relations below hold in U(U, Z):  �  p∈Tj,i  ηp  j,i ysp =  m  �  t=1  n  �  r=1  ωs  r,t xrj • yti  (12)  for all s = 1, · · · , m, i ∈ J and j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Furthermore, we can define a morphism of Lie g-modules ρU(U, Z): Z → U ⊗ U(U, Z) as  follows:  ρU(U, Z)(zr) :=  m  �  s=1  us ⊗ ysr,  for all r ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  (13)  It follows now that for all j ∈ I and i ∈ J we have:  ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z)(fj ↬ zi)  (10)  = ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='Z)  � �  p∈Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='i  ηp  ji zp  �  =  �  p∈Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='i  m  �  s=1  ηp  ji us ⊗ ysp =  m  �  s=1  �  us ⊗  �  p∈Tj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='i  ηp  ji ysp  �  (12)  =  m  �  s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='t=1  n  �  r=1  ωs  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='t us ⊗ xrj • yti =  m  �  t=1  n  �  r=1  � m  �  s=1  ωs  r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='t us  �  ⊗ xrj • yti  (9)  =  m  �  t=1  n  �  r=1  er ↷ ut ⊗ xrj • yti  (7)  =  m  �  t=1  fj ⇀ (ut ⊗ yti) = fj ⇀  m  �  t=1  ut ⊗ yti  (13)  = fj ⇀ ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z)(zi)  which shows that ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Z) is indeed a Lie g-modules map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  We will show that the pair  �  U(U, Z), ρU(U, Z)  �  constructed above is in fact the universal  A-module of U and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' To this end, consider a pair (X, f) consisting of an A-module X  and a morphism of Lie g-modules f : Z → U ⊗ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let {wsr | s = 1, · · · , m, r ∈ J} be a  family of elements of X such that for all r ∈ J we have:  g(zr) =  m  �  s=1  us ⊗ wsr  (14)  UNIVERSAL MODULES  7  Furthermore, as g: Z → U ⊗ X is a Lie g-modules map, a straightforward computation  shows that the following compatibilities hold for all s = 1, · · · , m, i ∈ J and j ∈ I:  �  p∈Tj,i  ηp  j,i wsp =  m  �  t=1  n  �  r=1  ωs  r,t xrj · wti  (15)  where · denotes the A-module action on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  The universal property of the free module yields a unique A-module map g: T (U, Z) → X  such that g(Ysr) = wsr, for all s = 1, · · · , m and r ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Moreover, Ker(g) contains the A-  submodule of T (U, Z) generated by the elements listed in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Indeed, as g : U(U, Z) →  X is a morphism of A-modules we have:  g  � �  p∈Tj,i  ηp  j,i Ysp −  m  �  t=1  n  �  r=1  ωs  r,t xrj • Yti  �  =  �  p∈Tj,i  ηp  j,i wsp −  m  �  t=1  n  �  r=1  ωs  r,t xrj · wti  (15)  = 0  for all s = 1, · · · , m, i ∈ J and j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' This shows that there exists a unique A-modules  map g: U(U, Z) → X such that g(ysr) = zsr, for all s = 1, · · · , m and r ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' This  implies that for all r ∈ J we have:  �  IdU ⊗ g  �  ρU(U, Z)(zr) =  �  IdU ⊗ g  �� m  �  s=1  us ⊗ ysr  �  =  m  �  s=1  us ⊗ wsr  (14)  = g(zr)  which means precisely that diagram (8) is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Moreover, g is obviously the  unique A-modules map with this property and the proof is now finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  □  The case g = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Particularizing the results of Section 2 for g = h, where h is the finite  dimensional Lie algebra defined in (1), leads to the following interesting consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  According to the discussion in Preliminaries, the universal algebra A(h, h) denoted by B  is in this case a bialgebra with coalgebra structure depicted in (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' This allows us to see  the tensor product U(U, Z) ⊗ U(U, Z) as well as the base field k as B-modules via the  comultiplication and the counit of B as follows:  xij ∗ (y ⊗ t) =  n  �  t=1  xit • y ⊗ xtj • t  (16)  xij · α = δijα  (17)  for all xij ∈ B, y, t ∈ U(U, Z) and α ∈ k, where • denotes the B-module strucuture on  U(U, Z) as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  First we show that if U is a finite dimensional Lie h-module as considered in (9), then  the B-module U(U, U) denoted by U(U) admits a coalgebra structure with respect to  which  �  U, ρU(U)  �  becomes a right U(U)-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let U be a finite dimensional Lie h-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' There exists a unique  coalgebra structure on U(U) such that  �  U, ρU(U)  �  becomes a right U(U)-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' In particular both U(U) ⊗ U(U) and k are B-modules via the formulas (16) and  (17) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Therefore, U ⊗ U(U) ⊗ U(U) and U ⊗ k are Lie h-modules via (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Furthermore, it can be easily checked that the maps  �  ρU(U) ⊗ IdU(U)  �  ρU(U) : U → U ⊗  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' AGORE  U(U) ⊗ U(U) and canU : U → U ⊗ k are morphisms of Lie h-modules, where canU : U →  U ⊗ k is the canonical isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Now Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='2 yields a unique B-modules map  ∆: U(U) → U(U) ⊗ U(U) such that the following diagram is commutative:  U  ρU(U)  �  �  ρU(U)⊗IdU(U)  �  ρU(U)  �❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  ❆  U ⊗ U(U)  IdU ⊗∆  �  U ⊗ U(U) ⊗ U(U)  Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' we obtain a unique B-modules map ε: U(U) → k such that the following  diagram is commutative:  U  ρU(U) �  canU  �■  ■  ■  ■  ■  ■  ■  ■  ■  ■  U ⊗ U(U)  IdU⊗ε  �  U ⊗ k  A straightforward computation shows that the commutativity of the two diagrams above  imply that ∆ and ε take the following form for all l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' t = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' m:  ∆(ylt) =  m  �  s=1  yls ⊗ yst,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  ε(ylt) = δlt1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  It is now obvious that  �  U(U), ∆, ε  �  form a coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Finally, by the commutativity of  the two diagrams above we obtain that  �  U, ρU(U)  �  is a right U(U)-comodule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' It is worth pointing out that with the coalgebra structure introduced  above, U(U) becomes a B-module coalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Indeed, having in mind that both ∆ and ε  are B-module maps, we have:  ∆(xab • ylt) = xab ∗ ∆(ylt) = xab ∗  � m  �  s=1  yls ⊗ yst  �(16)  =  n  �  c=1  m  �  s=1  xac • yls ⊗ xcb • yst  = (xab)(1) • (ylt)(1) ⊗ (xab)(2) • (ylt)(2)  and  ε(xab • ylt) = xab · ε(ylt)  (17)  = δab ε(ylt) = ε(xab) ε(ylt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  This shows that • is a coalgebra map, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  It turns out that the pair  �  U(U), ρU(U)  �  is universal in the following way:  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' For any coalgebra X with a B-module structure and any Lie h-module  morphism ψ: U → U ⊗X which makes U into a right X-comodule, there exists a unique  UNIVERSAL MODULES  9  B-modules and coalgebra morphism θ: U(U) → X such that the following diagram is  commutative:  U  ρU(U) �  ψ  �■  ■  ■  ■  ■  ■  ■  ■  ■  ■  U ⊗ U(U)  IdU ⊗θ  �  U ⊗ X  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' In light of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='2, such a unique A-modules map θ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' We are left to  show that θ is also a coalgebra map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' From the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='4 we know that θ is  defined for all l, t = 1, · · · , m by θ(ylt) = zlt where zlt are elements of X such that for all  r = 1, · · · , m we have ψ(ur) = �m  s=1 us ⊗ zsr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' As (U, ψ) is a right comodule, we obtain:  ∆(zlt) =  m  �  s=1  zls ⊗ zst,  ε(zlt) = δlt1k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  To this end, we have:  ∆  �  θ(ylt)  �  = ∆(zlt) =  m  �  s=1  zls ⊗ zst =  m  �  s=1  θ(yls) ⊗ θ(yst) = (θ ⊗ θ) ◦ ∆(ylt)  Similarly one can check that ε◦θ = ε which shows that θ is indeed a coalgebra map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The universal h-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' The second type of universal module one can consider  is the following:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Given an A-module V and a Lie g-module W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' the universal Lie h-  module of V and W is a pair  �  V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' W),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' τV(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' W )  �  consisting of a Lie h-module V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' W)  and a morphism of Lie g-modules τV(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' W ): W → V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' W)⊗V such that for any other pair  (Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' f) consisting of a Lie h-module Y and a morphism of Lie g-modules f : W → Y ⊗ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  there exists a unique morphism of Lie h-modules g: V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' W) → Y such that the following  diagram is commutative:  W  τV(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' W )  �  f  �❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  ❘  V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' W) ⊗ V  g⊗IdV  �  Y ⊗ V  (18)  The universal Lie h-module of V and W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' when it exists,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' can again be seen as the initial  object of the category whose objects are pairs (Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' f) consisting of a Lie h-module Y  and a morphism of Lie g-modules f : W → Y ⊗ V ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' while morphisms between two such  objects (Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' f) and (Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' f ′) are defined to be Lie h-module maps g: Y → Y ′ satisfying  (g ⊗ IdV ) ◦ f = f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let V be an A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Then, for all Lie g-modules W and all Lie  h-modules Y , we have a bijective correspondence between:  (1) Lie g-module maps f : W → Y ⊗ V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  (2) Lie h-module maps g: V(V, W) → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' AGORE  The universal h-module introduced in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='8 also exists provided that the A-  module V is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' If V is a finite dimensional A-module then the universal Lie h-module  of V and any other Lie g-module W exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' As this proof is somewhat similar in spirit with the one of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='4, we will be  brief and provide only the main ingredients required for the construction of the universal  Lie h-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Let {v1, · · · , vl}, l ∈ N∗, be a k-basis of the finite dimensional A-module V and denote  by γt  r,i,j ∈ k the structure constants of V with respect to its A-module structure ·, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  for all r = 1, · · · , n, i ∈ I and j = 1, · · · , l we have:  xri · vj =  l  �  s=1  γs  r,i,j vs  (19)  Consider {wr | r ∈ T} to be a k-basis for W and if ⊲ denotes its Lie g-module structure,  then for all j ∈ I and r ∈ T we can find a finite subset Sj,r of T such that:  fj ⊲ wr =  �  p∈Sj,r  σp  j,r wp  (20)  where σp  j,r ∈ k for all j ∈ I, r ∈ T, and p ∈ Sj,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Now let S(V, W) be the free Lie h-module on the set {Yri | r ∈ T, i = 1, · · · , l} and  denote by V(V, W) the quotient of S(V, W) by its Lie h-submodule generated by the  following elements:  �  p∈Sj,r  σp  j,r Yps −  l  �  k=1  n  �  p=1  γs  p,j,k ep ◮ Yrk  (21)  for all s = 1, · · · , l, r ∈ T and j ∈ I, where ◮ denotes the h-module action on S(V, W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  By denoting yri := �  Yri, where �  Yri stands for the equivalence class of Yri in the quotient  module V(V, W), it follows that the relations below hold in V(V, W):  �  p∈Sj,r  σp  j,r yps =  l  �  k=1  n  �  t=1  γs  t,j,k et ◮ yrk  (22)  for all s = 1, · · · , l, r ∈ T and j ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  It can now be easily seen, as in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='4, that the pair (V(V, W), τV(V, W ))  is the universal Lie h-module of V and W, where τV(V, W ): W → V(V, W) ⊗ V is the  morphism of Lie g-modules defined for all r ∈ T as follows:  τV(V, W )(wr) :=  l  �  s=1  yrs ⊗ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  (23)  □  UNIVERSAL MODULES  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Functors between module categories  In this section we show that the two universal module constructions previously introduced  are functorial and, moreover, if certain conditions are fulfilled the corresponding functors  admit right adjoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' We start, however, by stating the following easy consequence of  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let (U, ↷) ∈ hLM and (V, ·) ∈ AM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Then:  1) We have a functor U ⊗ −: AM → gLM from the category of A-modules to the  category of Lie g-modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  2) We have a functor − ⊗ V : hLM → gLM between the categories of Lie modules  over h and g respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' In light of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1, we are only left to show that morphisms behave well with  respect to the corresponding associative or Lie module structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' We will treat only the  first statement and leave the second one to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' To this end, consider (V, ·) and  (V ′, •) two A-modules, ⇀ and ⇀′ the corresponding induced Lie g-module actions via  (7) and g: V → V ′ a morphism in AM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Then, for all i ∈ I, l ∈ U and t ∈ V we have:  (IdU ⊗ g)  �  fi ⇀ (l ⊗ t)  �(7)  =  n  �  j=1  (ej ↷ l) ⊗ g(xji · t) =  n  �  j=1  (ej ↷ l) ⊗ xji • g(t)  (7)  = fi ⇀′ �  l ⊗ g(t)  �  □  We consider now the universal module functors:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let U be a finite dimensional Lie h-module and V a finite dimensional  A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  (1) There exists a functor UU : gLM → AM defined as follows for all Lie g-modules  X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Y and all morphisms f : X → Y in gLM:  UU(X) = U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  UU(f) = f  where f : U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' X) → U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Y ) is the unique A-modules morphism which makes  the following diagram commutative:  X  ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' X)  �  ρU(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Y )◦f  �◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  U ⊗ U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' X)  IdU⊗f  �  U ⊗ U(U,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Y )  (24)  (2) There exists a functor VV : gLM → hLM defined as follows for all Lie g-modules  X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Y and all morphisms f : X → Y in gLM:  VV (X) = V(V,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  VV (f) = f  12  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' AGORE  where f : V(V, X) → V(V, Y ) is the unique morphism of Lie h-modules which  makes the following diagram commutative:  X  τV(V, X)  �  τV(V, Y )◦f  �◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  V(V, X) ⊗ V  f⊗IdV  �  V(V, Y ) ⊗ V  (25)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' As the result follows in a straightforward manner by a standard category the-  ory argument, we only sketch the proof of the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Indeed, if f = IdX then  IdU(U, X) is obviously the unique A-modules morphism which makes diagram (24) com-  mute and therefore UU(IdX) = IdU(U, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Moreover, if f : X → Y and g: Y → W are two  morphisms in gLM, then g ◦f : U(U, X) → U(U, W) is obviously the unique A-modules  morphism which makes the following diagram commutative:  Z  ρU(U, X)  �  ρU(U, W )◦g◦f  �◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  ◗  U ⊗ U(U, X)  IdU⊗  �  g◦f  �  �  U ⊗ U(U, W)  and we can conclude that UU(g ◦ f) = UU(g) ◦ UU(f), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  □  Under the appropriate finite-dimensionality assumptions, the functors constructed in  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='1 are right adjoints to the universal module functors:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let (U, ↷) be a finite dimensional Lie h-module and (V, ·) a finite di-  mensional A-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Then:  1) The following functors form an adjunction:  UU : gLM → AM,  U ⊗ −: AM → gLM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  2) Similarly, the following functors also form an adjunction:  VV : gLM → hLM,  − ⊗ V : hLM → gLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' 1) As pointed out in Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='3, for all Lie g-modules Z and all A-modules  X, there is a bijection between HomAM  �  UU(Z), X  �  and HomgLM (Z, U ⊗ X) given as  follows for all morphisms of A-modules θ: UU(Z) → X:  ΓZ,X : HomAM (UU(Z), X) → HomgLM (Z, U ⊗ X),  ΓZ,X(θ) = (IdU ⊗ θ) ◦ ρU(U, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  The desired conclusion now follows by showing that the above bijection is natural in  both variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' This can be easily proved by a straightforward diagram chase and is left  to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  2) Using now Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='9, for all Lie g-modules W and all Lie h-modules Z, we obtain  a bijection between HomhLM  �  VV (W), Z  �  and HomgLM (W, Z ⊗ V ) defined as follows  UNIVERSAL MODULES  13  for all morphisms of Lie h-modules θ: VV (W) → Z:  ΓW,Z : HomhLM  �  VV (W), Z  �  → HomgLM (W, Z ⊗ V ),  ΓW,Z(θ) = (θ ⊗ IdV ) ◦ ρV(V, W ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  □  In particular, the two pairs of adjoint functors allow us to travel both ways between the  representation categories of the two (arbitrary) Lie algebras h and g and respectively  between the representation category of the associative algebra A and the representation  category of the Lie algebra g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Let ρi : g ⊗ Wi → Wi be representations of g, where i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' By the  colimit preservation property of left adjoints we can easily conclude that for any finite  dimensional Lie h-module U, UU(W1) ⊕ UU(W2) is the direct sum of the A-modules  UU(W1) and UU(W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  Similarly, for any finite dimensional A-module V , VV (W1) ⊕  UU(W2) is the direct sum of the Lie h-modules UU(W1) and UU(W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' This can be easily  extended to an arbitrary family of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
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+page_content=' - On quantum groups associated to a pair of preregular forms,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Noncommut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
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+page_content=', Lopez Franco, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=', Vasilakopoulou, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
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+page_content=' - Hopf Algebras, Benjamin New York, 1969.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
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+page_content=' - The coendomorphism bialgebra of an algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
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+page_content=' Tokyo Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' 37  (1990), 425–456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='  14  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' AGORE  Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium  Simion Stoilow Institute of Mathematics of the Romanian Academy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content=' Box 1-764, 014700  Bucharest, Romania  Email address: ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='agore@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NE1T4oBgHgl3EQfRQO7/content/2301.03051v1.pdf'}

1dAyT4oBgHgl3EQfPfaV/content/tmp_files/2301.00026v1.pdf.txt

@@ -0,0 +1,2071 @@
+Killing Horizons Decohere Quantum Superpositions
+Daine L. Danielson,1, ∗ Gautam Satishchandran,2, 1, † and Robert M. Wald1, ‡
+1Enrico Fermi Institute and Department of Physics,
+The University of Chicago, 933 East 56th Street, Chicago, Illinois 60637, USA
+2Princeton Gravity Initiative, Princeton University,
+Jadwin Hall, Washington Road, Princeton NJ 08544, USA
+(Dated: January 3, 2023)
+We recently showed that if a massive (or charged) body is put in a quantum spatial superposition, the
+mere presence of a black hole in its vicinity will eventually decohere the superposition. In this paper
+we show that, more generally, decoherence of stationary superpositions will occur in any spacetime
+with a Killing horizon. This occurs because, in effect, the long-range field of the body is registered
+on the Killing horizon which, we show, necessitates a flux of “soft horizon gravitons/photons”
+through the horizon. The Killing horizon thereby harvests “which path” information of quantum
+superpositions and will decohere any quantum superposition in a finite time. It is particularly
+instructive to analyze the case of a uniformly accelerating body in a quantum superposition in
+flat spacetime. As we show, from the Rindler perspective the superposition is decohered by “soft
+gravitons/photons” that propagate through the Rindler horizon with negligible (Rindler) energy.
+We show that this decoherence effect is distinct from—and larger than—the decoherence resulting
+from the presence of Unruh radiation. We further show that from the inertial perspective, the
+decoherence is due to the radiation of high frequency (inertial) gravitons/photons to null infinity.
+(The notion of gravitons/photons that propagate through the Rindler horizon is the same notion
+as that of gravitons/photons that propagate to null infinity.) We also analyze the decoherence of
+a spatial superposition due to the presence of a cosmological horizon in de Sitter spacetime. We
+provide estimates of the decoherence time for such quantum superpositions in both the Rindler and
+cosmological cases.
+1.
+INTRODUCTION
+Consider a stationary spacetime in which an experimen-
+talist, Alice, is present. Alice’s lab is stationary, and she
+has control of a charged or massive body (hereinafter re-
+ferred to as a “particle”). She sends her particle through a
+Stern-Gerlach apparatus or other device that puts her par-
+ticle in a quantum superposition of two spatially separated
+states1. She keeps these spatially separated components
+stationary for a time T and then recombines them. Will
+Alice be able to maintain the coherence of these compo-
+nents, so that, when recombined, the final state of her
+particle will be pure—or will decoherence have occurred,
+so that the final state of her particle will be mixed?
+Ordinarily, any decoherence effects will be dominated
+by “environmental influences,” i.e., additional degrees
+of freedom present in Alice’s lab that interact with her
+particle. We assume that Alice has perfect control of her
+laboratory and its environment so that there is no deco-
+herence from ordinary environmental effects. However,
+for a charged or massive particle, Alice cannot perfectly
+control the electromagnetic or gravitational field, since
+her particle acts as a source for these fields and some
+∗ daine@uchicago.edu
+† gautam.satish@princeton.edu
+‡ rmwa@uchicago.edu
+1 Quantum spatial superpositions of massive bodies have been of
+recent interest in both theoretical as well as proposed experimental
+probes of fundamental properties of quantum gravity, e.g., [1–13].
+radiation will be emitted during the portions of her ex-
+periment where she separates and recombines her particle.
+Nevertheless, in Minkowski spacetime, if her lab is sta-
+tionary in the ordinary, inertial sense, she can perform
+her experiment in a sufficiently adiabatic manner that
+negligible decohering radiation is emitted. In principle,
+she can keep the particle separated for an arbitrarily long
+time T and still maintain coherence when the components
+are recombined.
+In a recent paper [14], we showed that the above situ-
+ation changes dramatically if a black hole is present in
+the spacetime—even though the experiment is carried
+out entirely in the black hole’s exterior. In effect, a black
+hole horizon harvests “which path” information about any
+quantum superposition in its exterior, via the long-range
+fields sourced by the superposed matter. We showed that
+this results in the unavoidable radiation of entangling
+“soft photons or gravitons” through the horizon that carry
+the “which path” information into the black hole. Con-
+sequently, the mere presence of the black hole implies a
+fundamental rate of decoherence on the quantum super-
+position2. Although the rate of decoherence will be small
+if the black hole is far away, the coherence decays expo-
+nentially in the time, T, that the spatial superposition
+is maintained. Thus, in any spacetime with a black hole,
+there will be essentially complete decoherence within a
+2 In QED, this effect applies only to superpositions of charged particles.
+However, since all matter sources gravity, the quantum gravitational
+decoherence applies to all superpositions.
+arXiv:2301.00026v1  [hep-th]  30 Dec 2022
+
+2
+finite time3.
+The purpose of this paper is to generalize the results of
+[14] to spacetimes with Killing horizons, i.e., spacetimes
+with a Killing vector field such that there is a null surface
+to which the Killing field is normal (see, e.g., [15] for a
+discussion of properties of Killing horizons). The event
+horizon of a stationary black hole is a Killing horizon
+[16–18], so spacetimes with Killing horizons encompass
+the case of stationary spacetimes that contain black holes.
+However, there are many cases of interest where Killing
+horizons are present without the presence of black holes.
+One such case is that of Minkowski spacetime, where
+the Rindler horizon is a Killing horizon with respect to
+the Lorentz boost Killing field.
+Another such case is
+de Sitter spacetime, where the cosmological horizon is a
+Killing horizon. We will show that in these cases, a spatial
+superposition that is kept stationary (with respect to the
+symmetry generating the Killing horizon) will decohere
+in a manner similar to the black hole case. We will also
+provide an estimate of the maximum amount of time
+during which coherence can be maintained.
+The case of the Rindler horizon is particularly instruc-
+tive.
+The relevant symmetry here is that of Lorentz
+boosts, so Alice’s lab will be “stationary” if it is uniformly
+accelerating. Our analysis based upon radiation through
+the Rindler horizon shows that decoherence of a uniformly
+accelerating spatially separated superposition occurs be-
+cause of the emission of “soft” (i.e., very low frequency)
+gravitons or photons, where the frequency is defined rel-
+ative to an affine parameter on the Rindler horizon. As
+we shall show, the decoherence effect of this radiation of
+soft gravitons or photons is distinct from the (smaller)
+decoherence effect resulting from the presence of Unruh
+radiation. To gain further insight, we also analyze the
+decohering radiation in the electromagnetic case from the
+inertial point of view, using the Liénard-Wiechert solution
+to determine the radiation at future null infinity. As we
+shall show, the decohering photons are of high frequency
+at null infinity.
+In sec. 2 we provide a general discussion of the deco-
+herence of a quantum superposition due to radiation in a
+stationary spacetime. In sec. 3 we consider the decoher-
+ence of a uniformly accelerating superposition, analyzing
+it from both the Rindler and Minkowski viewpoints. We
+also show that this decoherence is distinct from (and larger
+than) the decoherence effects due to the presence of Un-
+ruh radiation. In sec. 4 we analyze the decoherence in de
+Sitter spacetime associated with the cosmological horizon.
+We will work in Planck units where G = c = ℏ = kB = 1
+and, in electromagnetic formulas, we also put ϵ0 = 1, but
+we will restore these constants in our formulas that give
+estimates for decoherence times. Lower case Latin indices
+represent abstract spacetime indices. Upper case Latin in-
+dices from the early alphabet correspond to spatial indices
+3 This maximal coherence time for superpositions in the exterior can
+be much smaller than the evaporation time of the black hole.
+on horizons or null infinity.
+2.
+DECOHERENCE DUE TO RADIATION IN A
+STATIONARY SPACETIME
+In this section, we will give a general analysis of the
+decoherence of a spatial superposition in a stationary
+spacetime due to emission of radiation by the body. Our
+analysis applies both to the decoherence of a charged
+body due to emission of electromagnetic radiation and to
+the decoherence of a gravitating body due to emission of
+linearized gravitational radiation. The analyses of these
+two cases are very closely parallel.
+In order to avoid
+repetition, we will analyze only the electromagnetic case
+in detail, but near the end of this section, we will state the
+corresponding results in the linearized gravitational case,
+which can be obtained straightforwardly by replacing the
+vector potential Aa with the perturbed metric hab, the
+charge-current ja with the stress-energy Tab, etc.
+Consider a charged particle4 in a stationary spacetime.
+We assume that the particle is initially in a stationary
+state. The particle is then put through a Stern-Gerlach (or
+other) apparatus, resulting in it being in a superposition
+state5
+|ψ⟩ =
+1
+√
+2 (|ψ1⟩ + |ψ2⟩)
+(2.1)
+where |ψ1⟩ and |ψ2⟩ are normalized states that are spa-
+tially separated after passing through the apparatus. The
+particle is then recombined via a reversing Stern-Gerlach
+(or other) apparatus and returns to a stationary state.
+We are particularly interested in the case where, between
+separation and recombination, |ψ1⟩ and |ψ2⟩ are kept
+stationary for a long period of time, T, but we do not
+make any such assumption in this section. We wish to
+estimate how much decoherence due to emission of elec-
+tromagnetic radiation will have occurred by the time of
+recombination6.
+4 As already indicated above, the “particle” need not be an elementary
+particle but could be a “nanoparticle” or any other body whose only
+relevant degree of freedom for our analysis is its center of mass.
+5 For simplicity, we have assumed that we have a 50-50 superposition
+of |ψ1⟩ and |ψ2⟩, but this assumption is not necessary.
+6 The decoherence of Alice’s particle can be experimentally deter-
+mined as follows. We assume that Alice’s particle initially has spin
+in the positive x-direction and thus is in a 50-50 superposition of
+z-spin after passing through the initial Stern-Gerlach apparatus.
+After recombination, Alice measures the x-spin of her particle. If
+coherence of the superposition eq. (2.1) has been maintained, then
+(assuming that Alice has made appropriate corrections if there are
+any phase differences between the paths) the spin will always be
+found to be in the positive x-direction. On the other hand, if any
+coherence has been lost, the particle will not be in a state of definite
+spin, and the spin will sometimes be found to be in the negative
+x-direction. By repeating the experiment many times, Alice can, in
+principle, determine the decoherence to any desired accuracy.
+
+3
+A key assumption that we shall make is that the fluctu-
+ations in the charge-current operator ja in the states |ψ1⟩
+and |ψ2⟩ are negligibly small over the scales of interest
+so that we can treat the charge current in each of these
+states as c-number sources in Maxwell’s equations, given
+by ja
+1 = ⟨ψ1|ja|ψ1⟩ and ja
+2 = ⟨ψ2|ja|ψ2⟩, respectively. In
+the initial and final stationary eras, |ψ1⟩ and |ψ2⟩ are
+assumed to coincide spatially (though they may differ in
+other characteristics, such as spin) so that ja
+1 = ja
+2 at very
+early and very late times.
+In order to proceed further, we must specify the initial
+state of the electromagnetic field. Since, prior to going
+through the Stern-Gerlach apparatus, the charge is as-
+sumed to be stationary, at early times we may subtract
+the “Coulomb field” Cin
+a of the charge, i.e., at early times
+we may consider the electromagnetic field observable
+Ain
+a = Aa − Cin
+a 1
+(2.2)
+where Cin
+a is the (assumed to be unique) stationary clas-
+sical solution to Maxwell’s equations with the early time
+stationary charged particle source ja
+1 = ja
+2 and Aa is
+the vector potential operator. We need not assume any
+specific choice of gauge for Ain
+a . Then Ain
+a satisfies the
+source-free Maxwell’s equations at early times, and we
+may extend its definition to all times by requiring it to
+satisfy the source-free Maxwell equations everywhere.
+The initial state of the electromagnetic field may be
+specified by giving the “radiation state” of Ain
+a .
+The
+choice of this state depends on the physical situation being
+considered. If the spacetime were globally stationary—i.e.,
+if the stationary Killing field were everywhere timelike, so,
+in particular, there are no Killing horizons—it would be
+natural to assume that the initial state of the radiation
+is the stationary vacuum state, i.e., the ground state
+relative to the time translations. For the case of a black
+hole spacetime, it would be correspondingly natural to
+assume that the initial state of the radiation is that of
+the Unruh vacuum, since for a black hole formed by
+gravitational collapse, the state of a quantum field is
+expected to approach the Unruh vacuum after the black
+hole has “settled down” to a stationary state. For the
+case of Minkowski spacetime, we take the initial state
+of the radiation to be the ordinary (inertial) Minkowski
+vacuum. For de Sitter spacetime, we take the initial state
+of the radiation to be the de Sitter invariant vacuum7 for
+the electromagnetic field [20]. We denote the initial state
+of the radiation in all of the above cases by |Ψ0⟩.
+In each of the above cases, |Ψ0⟩ is a pure, quasi-free (i.e.,
+Gaussian) state. It follows (see, e.g., [22] or appendix A
+of [15]) that we can construct a one-particle Hilbert space
+Hin and corresponding Fock space F(Hin) wherein |Ψ0⟩
+plays the role of the vacuum state and the field operator
+7
+A de Sitter invariant vacuum state does not exist for the massless
+scalar field [19] but such a state does exist for the electromagnetic
+field [20] and linearized gravitational field [21].
+Ain
+a is represented on F(Hin) by
+Ain
+a (f a) = ia(Kσf) − ia†(Kσf).
+(2.3)
+Here f a a divergence-free8 test function, σf denotes the
+advanced minus retarded solution to Maxwell’s equations
+with source f a, and K : S → Hin denotes the map taking
+the space S of classical solutions to their representatives
+in the one-particle Hilbert space Hin. The commutator
+of the creation and annihilation operators in eq. (2.3) is
+given by
+[a(Kσf), a†(Kσg)] = ⟨Kσf|Kσg⟩ 1.
+(2.4)
+where ��Kσf|Kσg⟩ is the inner product on Hin, which is
+given by a natural generalization of the Klein-Gordon
+inner product to electromagnetic fields.
+For the case of a globally stationary spacetime in the
+stationary vacuum state, Kσf corresponds to taking the
+positive frequency part of σf with respect to the time
+translations generating the stationary symmetry. For the
+case of a stationary black hole in the Unruh vacuum state,
+Kσf corresponds to taking the positive frequency part of
+σf with respect to affine time on the past horizon and
+with respect to Killing time at past null infinity. For
+Minkowski spacetime in the inertial Minkowski vacuum,
+Kσf corresponds to taking the positive frequency part
+of σf with respect to inertial time translations. Equiv-
+alently, Kσf, in this case, corresponds to the solution
+obtained by taking the positive frequency part of the re-
+striction of σf to any null hyperplane N (i.e., any Rindler
+horizon) with respect to an affine parametrization of the
+null geodesics generating N. For de Sitter spacetime in
+the de Sitter invariant vacuum, Kσf corresponds to the
+solution obtained by taking the positive frequency part
+of the restriction of σf to any cosmological horizon with
+respect to an affine parametrization of the null geodesics
+generating that horizon.
+Under the above assumption that the charge-current
+of |ψ1⟩ and |ψ2⟩ can be treated as c-number sources, the
+electromagnetic field Ai,a in the presence of the charge
+in state |ψi⟩ for i = 1, 2 is given in terms of the source
+free field Ain
+a by [23]
+Ai,a = Ain
+a + Gret
+a (jb
+i )1
+(2.5)
+where Gret
+a (jb
+i ) denotes the classical retarded solution for
+source jb
+i . In particular, since the field Ain
+a is in state
+|Ψ0⟩, the correlation functions of the electromagnetic field
+8 Restriction of the smearing to divergence-free test functions is
+necessary and sufficient to eliminate the gauge dependence of Ain
+a
+(see, e.g., P.101 of [22]).
+
+4
+Ai,a for |ψi⟩ are given by9
+⟨Ai,a1(x1) . . . Ai,an(xn)⟩
+= ⟨Ψ0|
+�
+Ain
+a1(x1) + Gret
+a1 (jb
+i )(x1)1)
+�
+. . .
+�
+Ain
+an(xn) + Gret
+an (jb
+i )(xn)1)
+�
+|Ψ0⟩.
+(2.6)
+Equation (2.6) is valid at all times.
+However, at
+late times—i.e., to the future of any Cauchy surface Σ
+corresponding to the time at which recombination has
+occurred—we can again subtract off the common sta-
+tionary Coulomb field, Cout
+a
+, of ja
+1 = ja
+2 to obtain the
+source-free field10 Aout
+i,a that describes the radiation at
+late times for the states |ψi⟩,
+Aout
+i,a = Ai,a − Cout
+a
+1 .
+(2.7)
+By eq. (2.6), at late times, the correlation functions of
+Aout
+a
+are given by
+⟨Aout
+i,a1(x1) . . . Aout
+i,an(xn)⟩
+= ⟨Ψ0|
+�
+Ain
+a1(x1) + Ai,a1(x1)1)
+�
+. . .
+�
+Ain
+an(xn) + Ai,an(xn)1)
+�
+|Ψ0⟩
+(2.8)
+where
+Ai,a = Gret
+a (jb
+i ) − Cout
+a
+.
+(2.9)
+Note that Ai,a is a classical solution of the source-free
+Maxwell equations in the late-time region.
+The correlation functions eq. (2.8) on any late-time
+Cauchy surface are precisely those of the coherent state
+|Ψi⟩ = e− 1
+2 ∥KAi∥2 exp
+�
+a†(KAi)
+�
+|Ψ0⟩ ,
+(2.10)
+where the norm is that of the one-particle inner product
+of eq. (2.4). Thus, the coherent state |Ψ1⟩ describes the
+“out” radiation state corresponding to charged particle
+state |ψ1⟩ and the coherent state |Ψ2⟩ describes the “out”
+radiation state corresponding to charged particle state
+|ψ2⟩. The joint “out” state, |Υ⟩, of the particle-radiation
+system is given by
+|Υ⟩ =
+1
+√
+2 (|ψ1⟩ ⊗ |Ψ1⟩ + |ψ2⟩ ⊗ |Ψ2⟩) .
+(2.11)
+Therefore, the decoherence of |ψ1⟩ and |ψ2⟩ due to emis-
+sion of electromagnetic radiation is given by
+D = 1 − | ⟨Ψ1|Ψ2⟩ |.
+(2.12)
+9 It is understood that each of the xk variables should be smeared
+with a divergence-free test vector field fa
+k .
+10Note that Ain
+a
+did not have a subscript “i” whereas Ai,a and
+Aout
+i,a do carry such subscripts. This is a consequence of the fact
+that we are working in the “in” representation—i.e., the Heisenberg
+representation on the Hilbert space F(Hin)—so Ain
+a does not depend
+on the sources, but the other fields do.
+We wish to evaluate D.
+By the general formula for the inner product of coherent
+states, we have
+| ⟨Ψ1|Ψ2⟩ | = exp
+�
+−1
+2||K(A1 − A2)||2
+�
+.
+(2.13)
+Now, in the late-time era, A1,a−A2,a is just the difference
+between the classical retarded solutions with sources ja
+1
+and ja
+2,
+A1,a −A2,a = Gret
+a (jb
+1)−Gret
+a (jb
+2) = Gret
+a (jb
+1 −jb
+2). (2.14)
+Consider the coherent state associated with Gret
+a (jb
+1 − jb
+2)
+in the late-time era. We refer to photons in this state as
+entangling photons. By the general properties of coherent
+states, the expected number, ⟨N⟩, of entangling photons
+is given by
+⟨N⟩ ≡ ||K
+�
+Gret(j1 − j2)
+�
+||2.
+(2.15)
+Thus, we have
+| ⟨Ψ1|Ψ2⟩ | = exp
+�
+−1
+2⟨N⟩
+�
+(2.16)
+so
+D = 1 − | ⟨Ψ1|Ψ2⟩ | = 1 − exp
+�
+−1
+2⟨N⟩
+�
+(2.17)
+and we see that the necessary and sufficient condition for
+significant decoherence (D ∼ 1) is ⟨N⟩ ≳ 1.
+We summarize the results that we have obtained above
+as follows. Under the assumptions we have made above,
+in order to calculate the decoherence, D, of the particle
+due to radiation, we carry out the following steps:
+(1) We obtain the expected charge current, ja
+1 and ja
+2,
+for the particle in states |ψ1⟩ and |ψ2⟩ of the super-
+position.
+(2) We
+calculate
+the
+classical
+retarded
+solution,
+Gret
+a (jb
+1 − jb
+2) for the difference of these charge cur-
+rents, which is a source-free solution at late times,
+since ja
+1 = ja
+2 at late times.
+(3) We calculate the one-particle state KGret(j1 − j2)
+corresponding to Gret
+a (jb
+1 − jb
+2) at late times.
+In
+the various cases, this corresponds to the follow-
+ing: (i) For a globally stationary spacetime initially
+in the stationary vacuum state, this one-particle
+state is the positive frequency part of the solution
+with respect to the time translations generating the
+stationary symmetry. (ii) For the case of a station-
+ary black hole initially in the Unruh vacuum, the
+one-particle state is the positive frequency part of
+the solution with respect to affine time on the past
+horizon and with respect to Killing time at past
+null infinity. (iii) For Minkowski spacetime initially
+in the Minkowski vacuum, the one-particle state
+is the positive frequency part of the solution with
+
+5
+respect to inertial time or, equivalently, the posi-
+tive frequency part with respect to affine time on
+any Rindler horizon. (iv) For de Sitter spacetime
+initially in the de Sitter invariant vacuum, the one-
+particle state is the positive frequency part of the
+solution with respect to affine time on any cosmo-
+logical horizon.
+(4) We compute the squared norm, ∥K[Gret(j1 −j2)]∥2,
+of this one-particle state at late times. This quan-
+tity is equal to the expected number of entangling
+photons, ⟨N⟩. The decoherence due to radiation is
+then given by
+D = 1 − exp
+�
+−1
+2∥K
+�
+Gret(j1 − j2)
+�
+∥2
+�
+.
+(2.18)
+As previously stated, the above analysis extends
+straightforwardly to the linearized gravitational case,
+where the perturbed metric, hab, is treated as a linear
+quantum field propagating in the background classical
+stationary spacetime. To compute the decoherence due
+to gravitational radiation in this case, we carry out the
+above steps, replacing Aa by hab and the charge-current
+ja by the stress-energy tensor Tab. The retarded solu-
+tion Gret
+a (jb) for Maxwell’s equations is replaced by the
+retarded solution Gret
+ab (Tcd) for the linearized Einstein
+equation. The map K : S → Hin is again obtained as
+in item (3) above and the inner product on Hin is again
+given by a natural generalization of the Klein-Gordon
+inner product to linearized gravitational fields. The de-
+coherence due to gravitational radiation is then given by
+the analog of eq. (2.18).
+The above analysis applies for any motion of the compo-
+nents of Alice’s superposition. We are primarily interested
+in the case where, during a time interval T1, Alice puts
+a particle of charge q (or mass m) into a spatial super-
+position, where the distance between the components of
+the particle wavefunction is d. She then keeps this super-
+position stationary in her lab for a time T. Finally, she
+recombines her particle over a time interval T2.
+In Minkowski spacetime in the case where Alice’s lab is
+inertial, Gret
+a (jb
+1 − jb
+2) will be nonzero at null infinity only
+at the retarded times corresponding to the time intervals
+T1 and T2. A rough estimate of the number of entangling
+photons was obtained in [3] using the Larmor formula for
+radiation in these eras, which, in natural units, yields
+⟨N⟩ ∼
+q2d2
+[min(T1, T2)]2
+(Minkowski, EM).
+(2.19)
+The corresponding result in the linearized gravitational
+case is [3]
+⟨N⟩ ∼
+m2d4
+[min(T1, T2)]4
+(Minkowski, GR).
+(2.20)
+Therefore, if Alice recombines her particle sufficiently
+slowly that T1, T2 ≫ qd in the electromagnetic case or
+T1, T2 ≫ md2 in the gravitational case, then she can main-
+tain the quantum coherence of her particle. In particular,
+Alice can keep the components of her particle separated
+for as long a time T as she likes without destruction of
+the coherence.
+As shown in [14], the situation is quite different if a
+black hole is present. In the electromagnetic case, even
+if T1, T2 ≫ qd so that a negligible number of entangling
+photons is emitted to infinity, there will be entangling
+radiation emitted into the black hole. For large T, the
+number of entangling photons increases with T as11
+⟨N⟩ ∼ M 3q2d2
+D6
+T
+(black hole, EM)
+(2.21)
+where M is the mass of the black hole, D is the proper
+distance of Alice’s lab from the horizon of the black hole,
+and we assume that D ≳ M. The corresponding result
+in the linearized gravitational case is
+⟨N⟩ ∼ M 5m2d4
+D10
+T
+(black hole, GR).
+(2.22)
+Thus, the coherence of Alice’s particle will always be
+destroyed within a finite time.
+In the next two sections, we will apply the above anal-
+ysis to the cases of Rindler spacetime and de Sitter space-
+time. Although we will explicitly analyze only the Rindler
+and de Sitter cases, it will be clear from our analysis of the
+next two sections—as well as our analysis in [14]—that it
+can be applied to any Killing horizon, provided only that
+the initial “vacuum state” |Ψ0⟩ of the electromagnetic
+and/or linearized gravitational field corresponds to one-
+particle states that are positive frequency with respect to
+affine time on the future Killing horizon.
+3.
+RINDLER HORIZONS DECOHERE
+QUANTUM SUPERPOSITIONS
+We now consider the case of Minkowski spacetime with
+Alice’s lab uniformly accelerating with acceleration a.
+Specifically, we take Alice’s lab to follow the orbit
+t = 1
+a sinh(aτ),
+z = 1
+a cosh(aτ)
+(3.1)
+of the boost Killing field
+ba = a
+�
+z
+� ∂
+∂t
+�a
++ t
+� ∂
+∂z
+�a�
+.
+(3.2)
+Here we have normalized ba such that baba = −1 on
+the worldline of Alice’s laboratory. Thus, ba is the four-
+velocity of Alice’s laboratory and τ is the proper time in
+11In the analysis of [14], we used the fact that the Unruh vacuum is
+well approximated by the Hartle-Hawking vacuum at low frequencies
+near the horizon of the black hole.
+
+6
+her lab. We introduce the null coordinates
+U ≡ t − z,
+V ≡ t + z
+(3.3)
+and the corresponding vector fields
+na ≡ (∂/∂V )a,
+ℓa ≡ (∂/∂U)a,
+(3.4)
+which are globally defined, future-directed null vector
+fields that satisfy ℓana = −1. In terms of these coordi-
+nates, the Minkowski spacetime metric is
+η = −dUdV + dx2 + dy2
+(3.5)
+and the boost vector field is given by
+ba = a
+�
+− Uℓa + V na�
+.
+(3.6)
+The boost Killing field is null on the two “Rindler hori-
+zons,” i.e., the two null planes U = 0 and V = 0, which
+divide Minkowski spacetime into four wedges. The orbits
+of the boost Killing field are future-directed and time-
+like within the “right Rindler wedge” WR which is the
+region U < 0 and V > 0. Thus, the “right Rindler wedge”
+WR—where Alice performs her experiment—is a static,
+globally hyperbolic spacetime where the notion of “time
+translations” is defined by Lorentz boosts.
+We refer to the null surface U = 0 as the future Rindler
+horizon and denote it as H +
+R . On the region V > 0 of
+H +
+R , it is useful to introduce the coordinate v by
+V = V0eav
+(3.7)
+where V0 is an arbitrary constant. Then, for V > 0 on
+H +
+R , we have
+ba��
+HR+ = aV
+� ∂
+∂V
+�a����
+HR+
+=
+� ∂
+∂v
+�a����
+HR+
+.
+(3.8)
+Since (∂/∂V )a on the horizon is tangent to the affinely
+parameterized null geodesic generators of H +
+R , we refer
+to V as the “affine time” on H +
+R , whereas we refer to v
+as the “boost Killing time” on H +
+R .
+1.
+Decoherence Due to Radiation of Soft
+Photons/Gravitons Through the Rindler Horizon
+We are now in position to apply the results of sec. 2
+to the Rindler case. We will first analyze the electromag-
+netic case and then give the corresponding results in the
+gravitational case.
+We assume that the electromagnetic field is initially
+in the Minkowski vacuum state. We assume that Alice
+possesses a charged particle that is initially stationary
+(with respect to the boost Killing field) in her (uniformly
+accelerating) lab. She then creates a quantum spatial
+superposition which is held stationary (with respect to
+the boost Killing field) for a proper time T and is then
+recombined. We wish to know the degree of decoherence
+of Alice’s particle due to emission of radiation. We may
+directly apply the analysis of sec. 2 to answer this question.
+The future Rindler horizon H +
+R (U = 0) does not meet
+the technical requirements of being a Cauchy surface for
+Minkowski spacetime, since there are inextendible time-
+like curves that remain in the past of H +
+R as well as
+inextendible timelike curves that lie in the future of H +
+R .
+However, as argued in [24], it is effectively a Cauchy sur-
+face for determining evolution of solutions to the wave
+equation. This is most easily seen in the conformally
+completed spacetime, where H +
+R is the past light cone of
+a point p ∈ I + except for the single generator that lies
+on I + and it also is the future light cone of a point on
+p′ ∈ I − except for the single generator that lies on I −.
+Data on the full past light cone of p would determine a
+solution to the past of H +
+R and data on the full future
+light cone of p′ would determine a solution to the future
+of H +
+R , thereby determining a solution everywhere in
+Minkowski spacetime. However, for solutions with ap-
+propriate decay, the data on the missing null geodesic
+generators of I + and I − can be determined by conti-
+nuity from the data on H +
+R . Consequently, data on H +
+R
+suffices to uniquely characterize solutions with appropri-
+ate decay. Consequently, the “out” states |Ψ1⟩ and |Ψ2⟩
+of the radiation are completely determined by data on
+H +
+R . Note that this contrasts sharply with the black hole
+case, where one would need data on both the future event
+horizon and future null infinity to characterize the “out”
+state of radiation.
+The decoherence of Alice’s particle due to radiation is
+given by eq. (2.17). In order to evaluate this, we first
+consider a classical point charge of charge q in the “right
+Rindler wedge” WR that is stationary with respect to the
+boost Killing field and lies at proper distance D from the
+bifurcation surface of the Rindler horizon. Such a charge
+will be uniformly accelerating with acceleration a given
+by
+a = 1
+D .
+(3.9)
+The explicit solution for such a stationary charge in the
+Rindler wedge has long been known [25–30]. The only
+nonvanishing component of the electromagnetic field in
+the region V > 0 of H +
+R is
+EU ≡ Fabℓanb =
+2a2q
+π(1 + a2ρ2)2
+(3.10)
+where ρ2 ≡ x2 + y2. Electromagnetic radiation through
+the Rindler horizon is described by the pullback, EA, of
+the electric field Ea = Fabnb to H +
+R , where the capital
+Latin indices from the early alphabet denote spatial com-
+ponents in the x and y directions. Since EA = 0 on the
+horizon for a uniformly accelerated charge, one may say
+that a charge held stationary in Alice’s lab does not pro-
+duce any radiation as determined on H +
+R —even though
+a uniformly accelerated charge radiates (inertial) energy
+
+7
+to future null infinity12.
+Now consider the case where the point charge is initially
+uniformly accelerating with acceleration a at a proper
+distance D = 1/a from the bifurcation surface of the
+Rindler horizon.
+The charge is then moved in the z-
+direction to a different orbit of the same boost Killing
+field, so that it has uniform acceleration a′ and lies at
+proper distance D′ = 1/a′ from the Rindler horizon. After
+the charge has reached its new location, the electric field
+on H +
+R is again given by eq. (3.10), but its value, E′
+U,
+will be different from its value at early times. Maxwell’s
+equations on H +
+R imply that
+DAEA = ∂V EU
+(3.11)
+where DA is the derivative operator on the R2 cross-
+sections of the horizon and capital Latin indices from
+the early alphabet are raised and lowered with the met-
+ric, δAB, on the cross sections. Eq. (3.11) implies that
+EA ̸= 0 whenever ∂V EU ̸= 0, so there will be radiation
+through the horizon as the charge is being moved. Most
+importantly, it implies that
+DA
+�
+�
+∞
+�
+−∞
+dV EA
+�
+� = ∆EU
+(3.12)
+where ∆EU = E′
+U −EU is the change in the radial electric
+field between the charge at positions D′ and D. Now, in
+a gauge where Aana = 0 on the horizon, the transverse
+(i.e., x-y) components of the electric field are related to
+the corresponding components of the vector potential by
+EA = −∂V AA.
+(3.13)
+Since the transverse components of the Coulomb field of a
+static charge vanish, we may replace the vector potential
+AA by the “Coulomb subtracted” vector potential AA
+defined by eq.(2.9), so we have
+EA = −∂V AA.
+(3.14)
+It then follows immediately from eq. (3.12) that the dif-
+ference, ∆AA, between the final and initial values of AA
+is given by
+DA(∆AA) = −∆EU
+(3.15)
+independently of the manner in which the charge is moved
+from D to D′. Equation (3.15) is an exact mathemati-
+cal analog of the electromagnetic memory effect at null
+infinity [31].
+12A uniformly accelerating charge has a nonvanishing inertial energy
+current flux Tabta through both H +
+R and I +, where ta denotes a
+Minkowski time translation. However, the flux of “boost energy”
+Tabba vanishes at both H +
+R and I +.
+For the explicit solution eq. (3.10), we have
+∆EU ≈ qda3(1 − a2ρ2)
+(1 + a2ρ2)3
+.
+(3.16)
+where d = D′ − D and we have assumed that
+d ≪ D = 1
+a .
+(3.17)
+From eq. (3.15), we find that ∆AA points in the ˆρ-
+direction and has magnitude
+|∆AA| = ∆Aρ ≈
+qda4ρ2
+(1 + a2ρ2)2 .
+(3.18)
+The key point is that even though EA = 0 at both late
+and early times, AA does return to its original value at
+late times, and the change, ∆AA, in the vector potential
+between late and early times is determined only by the
+initial and final positions of the charge.
+We now consider the quantized radiation through the
+horizon resulting from the displacement of the charge,
+assuming that, after the displacement, the charge is held
+at its new position, D′, forever.
+For the Fock space
+associated with the Minkowski vacuum state, the map K :
+S → Hin that associates one-particle states to classical
+solutions is given by taking the positive frequency part of
+the classical solution with respect to inertial time, with the
+inner product on Hin given by the Klein-Gordon product.
+For the electromagnetic field on H +
+R in a gauge where
+Aana on H +
+R , the “free data” on H +
+R is the pull-back,
+AA, of the vector potential. For two classical solutions
+with data A1,A and A2,A on H +
+R , the inner product of
+their corresponding one-particle states is given by [15, 32]
+⟨KA1| KA2⟩H +
+R = 2
+�
+R2
+dxdy
+∞
+�
+0
+ωdω
+2π δAB ˆ
+A1,A ˆ
+A2,B
+(3.19)
+where ˆ
+AA(ω, xB) is the Fourier transform of AA(V, xB)
+with respect to the affine parameter V . By the same
+reasoning as led to eq. (2.15), the expected number of
+photons on H +
+R in the coherent state associated to any
+classical solution AA is simply
+⟨N⟩ = ∥KA∥2
+H +
+R
+(3.20)
+where the norm is defined by the inner product eq. (3.19).
+However, since ∆AA
+̸=
+0, the Fourier transform,
+ˆ
+AA(ω, xB), of AA diverges as 1/ω as ω → 0.
+It fol-
+lows that the integrand of the expression for the norm
+given by the right side of eq. (3.19) also diverges as 1/ω as
+ω → 0, so the integral is logarithmically divergent. Thus,
+||KA||2
+H +
+R = ∞. Therefore, if Alice displaces a charged
+particle to a different orbit of the boost Killing field and
+the particle remains on this new uniformly accelerated
+trajectory forever, an infinite number of “soft horizon
+
+8
+photons” will be radiated through the Rindler horizon
+regardless of how quickly or slowly this process is done.
+This is an exact mathematical analog of the infrared di-
+vergences that occur at null infinity in QED for processes
+with nonzero memory (see e.g., [33–35]).
+Now suppose that Alice displaces the particle a z-
+distance d ≪ D = 1/a from D to D′ = D+d as above, but
+instead of leaving the particle at D′ forever, she leaves it
+there for proper time13 T and then returns it to D. In this
+case, the transverse components of the vector potential,
+AA, return to their initial values at late times, so there
+is no “memory effect” at the horizon. Correspondingly,
+there are no infrared divergences in the expected number
+of photons that propagate through H +
+R . Nevertheless, if
+T is very large then the expected number of photons ⟨N⟩
+will be correspondingly large. To see this, we note that
+if, for convenience, we work in a gauge where AA = 0
+initially, then during the era at which the particle is at D′,
+AA will be given by the right side of eq. (3.18). If we keep
+the manner in which the particle is moved from D to D′
+as well as from D′ to D fixed but take T to be very large,
+the asymptotic behavior of the norm eq. (3.19) will be
+dominated by the low-frequency contribution from the era
+of time T that the particle is displaced. The logarithmic
+divergence at ω = 0 that would occur if the particle re-
+mained at D′ forever is now effectively cut off at frequency
+ω ∼ 1/V , where V denotes the affine time duration on
+the horizon H +
+R over which the particle remains at D′.
+We obtain
+⟨N⟩ = ||KA||2
+HR ∼ q2d2a2 ln
+�
+V
+min[V1, V2]
+�
+(3.21)
+where V1, V2 ≪ V are the durations of affine time over
+which the particle is displaced from D to D′ and from
+D′ back to D, so that 1/min[V1, V2] provides an effective
+high-frequency cutoff. However, the affine time V on the
+horizon is related to boost Killing time on the horizon by
+V = V0 exp(av)
+(3.22)
+and the boost Killing time v corresponds to the proper
+time T in Alice’s lab. Thus, we obtain
+⟨N⟩ ∼ q2d2a3T
+(Rindler, EM) .
+(3.23)
+Therefore, no matter how slowly the particle is displaced,
+it is forced to radiate a number of “soft Rindler horizon
+photons” through the Rindler horizon that is proportional
+to the time T that the particle remains on the displaced
+trajectory.
+We are now in a position to fully analyze Alice’s exper-
+iment. Alice’s lab is uniformly accelerating with acceler-
+13We have normalized the boost Killing field ba so that Killing time
+equals proper time on the orbit at D with acceleration a. Since we
+assume d = D′ − D ≪ D, Killing time and proper time are also
+(nearly) equal on the orbit at D′. Thus, T is also the elapsed Killing
+time that Alice keeps the particle at D′.
+ation a in Minkowski spacetime. She puts her particle
+of charge q into a superposition of states separated by
+z-distance d ≪ 1/a and keeps these components sta-
+tionary in her lab for a proper time T.
+She then re-
+combines the components and determines their coher-
+ence14. By the analysis of sec. 2, the decoherence is given
+by eq. (2.18). However, for large T, the calculation of
+||K [Gret(j1 − j2)] ||2 corresponds precisely to the calcu-
+lation we have given above of the number of photons
+radiated through the Rindler horizon when a charge is
+displaced for a time T. Thus, we obtain
+||K
+�
+Gret(j1 − j2)
+�
+||2 ∼ q2d2a3T.
+(3.24)
+In other words, for large T, Alice’s superposition will de-
+cohere due to radiation of “soft Rindler horizon photons,”
+as
+D = 1 − exp(−ΓradT)
+(3.25)
+where the “decoherence rate” Γrad, is given by,
+Γrad = q2d2a3.
+(3.26)
+Thus, restoring the constants c, ℏ, and ϵ0, Alice’s par-
+ticle will decohere within a time
+TD ∼ ϵ0ℏc6
+a3q2d2
+(Rindler, EM)
+(3.27)
+∼ 1033 years
+�g
+a
+�3
+·
+�e
+q
+�2
+·
+�m
+d
+�2
+.
+(3.28)
+Thus, if Alice’s lab uniformly accelerates at one g in
+flat spacetime and she separates an electron into two
+components one meter apart, she would not be able to
+maintain coherence of the electron for more than 1033
+years.
+A similar analysis holds in the gravitational case15
+where Alice separates a massive body with mass m across
+a distance d and maintains this superposition for a time
+T. In the gravitational case, the “electric part” of the
+perturbed Weyl tensor Eab = Cacbdncnd plays an analo-
+gous role to the electric field Ea in the electromagnetic
+version of the gedankenexperiment. For a uniformly ac-
+celerating point mass, the only non-vanishing compo-
+nent of the electric part of the Weyl tensor on H +
+R is
+EUU = Cacbdℓancℓbnd.
+Gravitational radiation on the horizon is described
+by the pullback, EAB, of Eab, which vanishes for the
+static point mass. However, the process of quasistatically
+moving the static point mass involves a change in EUU
+on H +
+R . The (once-contracted) Bianchi identity on the
+14The coherence can be determined as described in footnote 6.
+15In the gravitational case, additional stress-energy will be needed
+to keep Alice’s particle in uniform acceleration. We will ignore the
+gravitational effects of this additional stress-energy.
+
+9
+horizon yields
+DAEAB = ∂V EUB,
+DAEUA = ∂V EUU
+(3.29)
+which implies that
+DADBEAB = ∂2
+V EUU
+(3.30)
+which is closely analogous to eq. (3.11). As in the elec-
+tromagnetic case, if a uniformly accelerating point mass
+is quasistatically moved there is necessarily gravitational
+radiation through H +
+R .
+To determine the number of “Rindler horizon gravitons”
+emitted we quantize the linearized gravitational field. For
+a metric perturbation hab in a gauge where habna = 0
+and δABhAB = 0, the “free data” on H +
+R
+is hAB. A
+“particle” in the standard Fock space associated to the
+Poincaré invariant vacuum is then a positive frequency
+solution with respect to affine parameter V and the inner
+product on the one-particle Hilbert space is given by a
+direct analog of eq. (3.19) with the vector potential AA
+replaced with the metric perturbation hAB, namely
+⟨Kh1| Kh2⟩H +
+R = 1
+8
+�
+R2
+dxdy
+∞
+�
+0
+ωdω
+2π δABδCDˆh1,ACˆh2,BD.
+(3.31)
+Finally, EAB is related to the metric perturbation hAB
+by
+EAB = −1
+2∂2
+V hAB .
+(3.32)
+Equations (3.30) and (3.32) directly imply that a per-
+manent change, ∆EUU ̸= 0, in the U-U component of
+the electric part of the Weyl tensor on H +
+R
+implies a
+permanent change, ∆hAB ̸= 0, in the perturbed metric
+on H +
+R between early and late times. In the quantum
+theory, as in the electromagnetic case, this implies a log-
+arithmic infrared divergence in the number of gravitons
+emitted through H +
+R in the process where a uniformly
+accelerating charge is moved to a new orbit of the same
+boost Killing field and then remains at the new position
+forever.
+The analysis of Alice’s experiment proceeds in a similar
+manner to the electromagnetic case. Alice does not main-
+tain the relative separation of her wavefunction forever
+but closes her superposition after a proper time T. As
+before, the number of entangling gravitons emitted to
+the Rindler horizon is logarithmically growing in affine
+time and therefore linearly growing in the proper time
+duration T of Alice’s experiment. We obtain
+⟨N⟩ ∼ m2d4a5T
+(Rindler, GR) .
+(3.33)
+Thus, restoring constants, we find that the Rindler hori-
+zon decoheres the quantum superposition of a uniformly
+accelerating massive body in a time
+T GR
+D
+∼
+ℏc10
+Gm2d4a5
+(Rindler, GR)
+(3.34)
+∼ 2 fs
+�MMoon
+m
+�2
+·
+�RMoon
+d
+�4
+·
+�g
+a
+�5
+.
+(3.35)
+Therefore, if the Moon were accelerating at one g and
+occupied a quantum state with its center of mass super-
+posed by a spatial separation of the order of its own radius
+then it would decohere within about 2 femtoseconds. Of
+course, it would not be easy to put the moon in such a
+coherent quantum superposition.
+Note the acceleration of a stationary observer outside
+of a black hole who is reasonably far16 (D ≳ M) from the
+event horizon is a ∼ M/D2. If we substitute a = M/D2
+in eqs. (3.27) and (3.34), we obtain eqs. (2.21) and (2.22),
+respectively. Therefore, it might be tempting to believe
+that what is important in all cases is the acceleration of
+Alice’s lab. However, this is not the case. In particular,
+if we replace the black hole by an ordinary star (and if
+there are no dissipative effects in the star), then there
+will not be any analogous decoherence effect, even though
+the acceleration of Alice’s lab is the same as in the case
+of a black hole. Furthermore, as we shall see in sec. 4,
+decoherence effects associated with the cosmological hori-
+zon occur in de Sitter spacetime even for nonaccelerating
+observers. It is the presence of a Killing horizon that
+is the essential ingredient for the fundamental rate of
+decoherence of quantum superpositions as described in
+this paper.
+We now consider another potential cause of decoherence,
+namely Unruh radiation.
+2.
+Decoherence Due to Scattering of Unruh
+Radiation
+The Minkowski vacuum state restricted to a Rindler
+wedge is a thermal state at the Unruh temperature
+T = a
+2π
+(3.36)
+relative to the notion of time translations defined by
+the Lorentz boost Killing field ba, eq. (3.2). Thus, the
+superposition state of Alice’s particle will be buffeted by
+this thermal bath of Unruh radiation. Scattering of this
+radiation will cause some decoherence of Alice’s particle.
+Indeed, since this decoherence should occur at a steady
+rate while the superposition is kept stationary (and thus
+the decoherence will be proportional to T), one might even
+16It should be emphasized that the estimates made in [14] that yielded
+eqs.(2.21) and (2.22) assumed that Alice’s lab is reasonably far from
+the black hole. If Alice’s lab is extremely close to the black hole
+(i.e., at a distance D ≪ M from the horizon), then the black hole
+analysis would reduce to the Rindler case analyzed here.
+
+10
+suspect that scattering of Unruh radiation could be the
+same effect as found in the previous section but expressed
+in a different language. The purpose of this subsection
+is to show that this is not the case, i.e., decoherence
+due to scattering of Unruh radiation and decoherence
+due to radiation of “soft” photons/gravitons through the
+horizon are distinct effects. Furthermore, we shall show
+that, for reasonable parameter choices, the decoherence
+rate due to the scattering of Unruh radiation is smaller
+than the decoherence rate due to emitted radiation as
+obtained in the previous section. We will consider only
+the electromagnetic case in this subsection.
+The decoherence rate of a spatial superposition due
+to collisions with particles in an environment has been
+analyzed in [36–39], and we will adapt this analysis to
+obtain a rough estimate of the decoherence caused by the
+scattering of Unruh radiation. As in eq. (2.1), Alice has
+a particle of charge q in a state |ψ⟩ = (|ψ1⟩ + |ψ2⟩)/
+√
+2,
+where |ψ1⟩ and |ψ2⟩ are spatially separated by a distance
+d. Since we require d ≪ 1/a (see eq. (3.17)) and since
+the typical wavelength of Unruh photons at temperature
+eq. (3.36) is λ ∼ 1/a, we are in the scattering regime
+where λ ≫ d.
+In an elastic scattering event between
+Alice’s particle and a photon in the Unruh radiation, the
+final outgoing state of the photon will depend upon which
+branch of the superposition the photon scattered off of.
+Let |χ1⟩ denote the outgoing state of the Unruh photon
+for scattering off of |ψ1⟩ and let |χ2⟩ denote the outgoing
+state for scattering off of |ψ2⟩. Decoherence will occur to
+the extent to which these outgoing states of the scattered
+Unruh photon are distinguishable, i.e., D = 1−| ⟨χ1|χ2⟩ |.
+In order to obtain a rough estimate of the decoherence
+resulting from a single scattering event, we consider the
+corresponding Minkowski process of the scattering of a
+photon of momentum p off of an inertial superposition
+separated by d, with d ≪ 1/p. Assuming that the charged
+particle states |ψ1⟩ and |ψ2⟩ are identical except for their
+location, the scattered photon states |χ1⟩ and |χ2⟩ should
+differ only by the action of the translation operator e−i ⃗P·⃗d,
+i.e.,
+|χ2⟩ ≈ e−i ⃗P·⃗d |χ1⟩
+(3.37)
+where ⃗P denotes the photon momentum operator. Ex-
+panding the exponential, we obtain the following rough
+estimate of the decoherence resulting from a single scat-
+tering event involving a photon of momentum p
+1 − | ⟨χ1|χ2⟩ | ∼ p2d2
+(3.38)
+where we have ignored any dependence on the angle be-
+tween the incoming momentum ⃗p and the separation ⃗d.
+We will take eq. (3.38) as our estimate of the decoherence
+of Alice’s particle resulting from the scattering of a single
+Unruh photon of “Rindler momentum” p (i.e., of energy
+ϵ = p with respect to the boost Killing field ba).
+The total decoherence rate due to scattering of Unruh
+radiation is then given by
+Γscatt ∼ d2
+∞
+�
+0
+dp p2ϱ(p)σ(p)
+(3.39)
+where ϱ(p) is the number density of photons at momentum
+p (so ϱ(p) is also the incoming flux of photons) and σ(p)
+is the scattering cross-section. For a thermal distribution
+of photons17 we have
+ϱ(p) ∼
+p2
+ep/T − 1.
+(3.40)
+We take σ to be given by the Thomson cross-section
+σ = 8π
+3
+q4
+(4πm)2 ,
+(3.41)
+where m is the mass of Alice’s particle. Putting this all
+together, our estimate of the decoherence rate due to
+scattering of Unruh photons is
+Γscatt ∼ q4d2a5
+m2
+(Rindler, EM) .
+(3.42)
+Comparing eq. (3.42) to the rate of decoherence, Γrad
+due to the emission of soft photons given by eq. (3.26),
+one can immediately see that the effects are distinct.
+In particular, Γrad has no dependence on the mass, m,
+of Alice’s particle, whereas Γscatt does depend on m on
+account of the mass dependence of the scattering cross-
+section. The ratio of these decoherence rates is given
+by
+Γscatt
+Γrad
+∼ q2a2
+m2 =
+�q/m
+D
+�2
+(3.43)
+Now, q/m is the “charge radius” of Alice’s particle and,
+as argued in [3], it represents a fundamental lower bound
+to the spread of a charged particle due to vacuum fluc-
+tuations of the electromagnetic field. Therefore, in order
+that |ψ1⟩ and |ψ2⟩ not overlap, we must have d > q/m.
+Since d ≪ D, we conclude that
+Γscatt
+Γrad
+≪ 1
+(3.44)
+i.e., the contribution to decoherence from the scattering
+of Unruh radiation is negligible compared with the de-
+coherence due to emission of soft photons through the
+Rindler horizon.
+A similar analysis holds for a charged particle superpo-
+sition outside of a black hole. It is worth noting, that the
+17The factor of p2 in the numerator of eq. (3.40) arises from the density
+of states in Minkowski spacetime. We ignore here any differences
+between the Minkowski and Rindler densities of states.
+
+11
+decoherence effects due to scattering of Hawking radiation
+will decrease with distance, D, from the black hole only
+as 1/D2 for large D, giving,
+Γscatt ∼
+q4d2
+m2M 3
+1
+D2
+(black hole, EM).
+(3.45)
+On the other hand, by eq. (2.21) the decoherence effects
+of radiation of soft photons through the horizon decreases
+with D as 1/D6. Thus at sufficiently large D, the deco-
+herence effects due to scattering of Hawking radiation
+will dominate. However, in this regime, both effects are
+extremely small.
+3.
+Decoherence From the Inertial Perspective
+In our analysis of the decoherence of a spatial superpo-
+sition in the presence of a black hole [14] as well as in our
+analysis of the decoherence of a spatial superposition in
+Rindler spacetime given above in sec. 3.1, it may appear
+that we have introduced a radical new mechanism for de-
+coherence, namely radiation of soft photons and gravitons
+through a horizon. The main purpose of this subsection
+is to show that, in fact, the decoherence we derived in the
+Rindler case can also be obtained by entirely conventional
+means. In the Rindler case, we are simply considering a
+uniformly accelerating superposition in Minkowski space-
+time. The radiation of entangling photons to infinity from
+such a superposition can be calculated in the inertial view-
+point by standard methods, without introducing concepts
+such as a Rindler horizon. It is instructive to calculate
+the decoherence from the inertial viewpoint both in order
+to validate the results of sec. 3.1 as well as to gain insight
+into how the emitted “soft photons” would be interpreted
+by an inertial observer. As we shall see, the entangling
+photons as seen by inertial observer at large distances
+near θ = 0 will be “hard” even though, from her point of
+view, Alice has performed the experiment adiabatically.
+We will restrict our analysis in this subsection to the
+electromagnetic case.
+The Liénard-Wiechert solution for the potential of a
+point charge in Minkowski spacetime following an arbi-
+trary worldline Xµ(τ) is, in Lorenz gauge,
+Aµ(x) = 1
+4π
+1
+α
+q
+|⃗x − ⃗X(tret)|
+dXµ
+dt (tret)
+(3.46)
+where
+α ≡ 1 − ˆn · d ⃗X
+dt (tret)
+and ˆn = ⃗x − ⃗X(tret)
+|⃗x − ⃗X(tret)|
+.
+(3.47)
+For a uniformly accelerated trajectory with acceleration
+a, we have
+Xµ(τ) =
+�1
+a sinh(aτ), 0, 0, 1
+a cosh(aτ)
+�
+.
+(3.48)
+In Bondi coordinates (u, r, θ, φ) with
+u ≡ t − r
+(3.49)
+the future light cone of an event at proper time τ on the
+worldline eq. (3.48) reaches null infinity at
+au = sinh(aτ) − cos θ cosh(aτ).
+(3.50)
+Electromagnetic radiation is described by the pullback
+of the electromagnetic field, eq. (3.46), to null infinity.
+Taking the limit as r → ∞ at fixed u, we obtain18
+AA(u, θ, φ) = −q
+4π
+sinh(aτ) sin θ
+cosh(aτ) − cos θ sinh(aτ)(dθ)A
+(3.51)
+where, in this subsection, capital indices from the early
+alphabet denote angular components on the 2-sphere cross-
+sections of I +. We will be concerned with the difference,
+at fixed (u, θ, φ), between the electromagnetic radiation
+of a particle following the trajectory eq. (3.48) and a
+particle following a similar trajectory that is displaced in
+the z-direction by a proper distance d ≪ 1/a and thus
+has
+δa = a2d.
+(3.52)
+We denote this difference by
+Ad
+A(u, θ, φ) ≡ AA(a + δa) − AA(a) ≈ δa
+�∂AA
+∂a
+�
+u,θ
+(3.53)
+From eq. (3.51), we obtain
+Ad
+A = −a2qd
+4π
+u sin θ
+(cosh(aτ) − cos θ sinh(aτ))3 (dθ)A
+(3.54)
+where eq. (3.50) was used to compute (∂τ/∂a)(u,θ).
+In her experiment, Alice starts with her particle in a
+uniformly accelerating state. Over a proper time T1, she
+separates it into two uniformly accelerating components
+separated by a distance d as above.
+She keeps these
+components separated for a proper time T, and she then
+recombines them over a proper time T2. The difference
+between the radiation fields of these components is given
+by
+AA ≡ A1,A − A2,A = F(τ)Ad
+A
+(3.55)
+where the smooth function F is such that F(τ) = 0 for
+τ < −T1 and τ > T +T2, whereas F(τ) = 1 for 0 < τ < T.
+18The vector potential is not smooth at I + in Lorenz gauge but
+one can do an asymptotic gauge transformation such that Aa is
+smooth at I +. Such a gauge transformation does not affect the
+angular components AA at I + [35], so we can calculate AA using
+our Lorenz gauge expression.
+
+12
+The entangling photon content is then given by
+⟨N⟩ = ||KA||2 = 2
+�
+S2
+dΩ
+∞
+�
+0
+ωdω
+2π
+ˆ
+AA ˆ
+AA
+(3.56)
+where
+ˆ
+AA(ω, θ, φ) denotes the Fourier transform of
+AA(u, θ, φ) with respect to u, i.e.,
+ˆ
+AA(ω, θ, φ) =
+∞
+�
+−∞
+du eiωuAA(u, θ, φ).
+(3.57)
+We are interested in estimating ⟨N⟩ for large T.
+In order to evaluate the Fourier transform integral, it
+is useful to note that, at fixed a, we have
+du
+dτ = cosh(aτ) − cos θ sinh(aτ)
+(3.58)
+and
+d2u
+dτ 2 = a2u.
+(3.59)
+It follows that
+d
+du
+�
+1
+du/dτ
+�
+=
+1
+du/dτ
+d
+dτ
+�
+1
+du/dτ
+�
+=
+−a2u
+(cosh(aτ) − cos θ sinh(aτ))3
+(3.60)
+Thus, we have
+Ad
+A = qd sin θ
+4π
+(dθ)A
+d
+du
+�
+1
+du/dτ
+�
+(3.61)
+and
+ˆ
+AA = qd sin θ
+4π
+(dθ)A
+∞
+�
+−∞
+du eiωuF(τ) d
+du
+�
+1
+du/dτ
+�
+.
+(3.62)
+Integrating by parts, we obtain
+ˆ
+AA(ω, xA) = − qd sin θ
+4π
+(dθ)A
+�
+iω
+∞
+�
+−∞
+du eiωu F(τ)
+du/dτ
++
+∞
+�
+−∞
+du eiωu
+F ′(τ)
+(du/dτ)2
+�
+.
+(3.63)
+The second term in this equation contributes only during
+the time intervals (−T1, 0) and (T, T + T2) when Alice
+opens and closes the superposition. For large T, its con-
+tribution can be shown to be negligible compared with
+the first term. Therefore, we have
+ˆ
+AA(ω, xA) ≈ −(dθ)A
+iωqd sin θ
+4π
+I
+(3.64)
+where
+I ≡
+∞
+�
+−∞
+du eiωu F(τ)
+du/dτ .
+(3.65)
+To evaluate I, we approximate F by a step function in
+the τ-interval [0, T]. The corresponding interval, [u0, uT ],
+in u is
+u0 = −1
+a cos θ
+uT = 1
+2a
+�
+eaT (1 − cos θ) − e−aT (1 + cos θ)
+�
+.
+(3.66)
+Noting that
+du
+dτ =
+�
+a2u2 + sin2 θ
+(3.67)
+we obtain
+I ≈
+uT
+�
+u0
+du
+eiωu
+�
+a2u2 + sin2 θ
+.
+(3.68)
+It can be seen that for large T, the dominant contribution
+to I will come from small angles, θ ≪ 1. For aT ≫ 1, the
+upper limit of the integral may then be approximated as
+uT ≈ 1
+4aeaT θ2 − 1
+ae−aT
+for θ ≪ 1
+∼
+�
+0
+for θ2/4 < e−aT
+1
+4aθ2eaT
+for θ2/4 ≥ e−aT .
+(3.69)
+For aT ≫ 1, the contribution to I from θ2/4 < e−aT
+can be shown to make a negligible contribution to ⟨N⟩,
+eq. (3.56). Therefore, we may approximate I as
+I ∼ Θ(θ2 − 4e−aT )
+exp(aT )θ2/(4a)
+�
+−1/a
+du
+eiωu
+�
+a2u2 + sin2 θ
+(3.70)
+where
+Θ(x) ≡
+�
+0
+for x < 0
+1
+for x ≥ 0.
+(3.71)
+For 0 < ω < 4ae−aT /θ2, we may bound I by replacing
+eiωu by 1. The integral can then be evaluated explic-
+itly, and it can be shown that for aT ≫ 1, the con-
+tribution to ⟨N⟩ from this frequency range is negligi-
+ble. For ω > 4ae−aT /θ2, the integrand is oscillatory for
+u > exp(aT)θ2/(4a), and, for aT ≫ 1, we will make neg-
+ligible error in our estimate of ⟨N⟩ if we replace the upper
+limit of eq. (3.70) by ∞. We will also make a negligible
+error by replacing the lower limit by 0. Thus, for aT ≫ 1,
+
+13
+we may approximate I as
+I ∼ Θ(θ2−4e−aT )Θ(ω−4ae−aT /θ2)
+∞
+�
+0
+du
+eiωu
+�
+a2u2 + sin2 θ
+.
+(3.72)
+Evaluating the integral we obtain
+I ∼ 1
+aΘ(θ2 − 4e−aT )Θ(ω − 4ae−aT /θ2)
+�1
+2iπI0(sin θω/a)
++K0(sin θω/a) − 1
+2iπLLL0(sin θω/a)
+�
+(3.73)
+where I0, K0 are Bessel functions and LLL0 is a Struve
+function. This expression is highly suppressed for ω > a/θ,
+so we can expand in θω/a and truncate the function above
+ω = a/θ to obtain,
+I ∼ −1
+aΘ(1−θω/a)Θ(θ2−4e−aT )Θ(ω−4ae−aT /θ2) ln (θω/a) .
+(3.74)
+Note that the restrictions ω < a/θ, and θ > 2e−aT/2 im-
+ply a frequency cutoff at ω ∼ aeaT/2/2. By eqs.(3.74) and
+(3.64), the frequency spectrum of ˆ
+AA goes as ω ln(ω/a)
+up to this cutoff, i.e., the spectrum is “hard” and becomes
+increasingly so for large T. This contrasts with the in-
+creasingly “soft” spectrum on the Rindler horizon, which
+goes as 1/ω down to a low frequency cutoff ∼ 1/V ∝ e−aT .
+Thus, the “soft horizon photons” from the Rindler per-
+spective are “hard” photons from the inertial perspective.
+From eq. (3.56) for ⟨N⟩ together with our expression
+eq. (3.64) for ˆ
+AA and the expression eq. (3.74) that we
+have just derived for I, we obtain
+⟨N⟩ ∼
+�qd
+a
+�2 �
+dωdθ θ3ω3
+�
+ln ωθ
+a
+�2
+(3.75)
+where the region of ω-θ integration is determined by the Θ-
+functions appearing in eq. (3.74) as well as the geometrical
+restriction θ ≲ 1. We can break up this region into the
+portion with ω ≤ a and the portion with ω > a. Since
+the region with ω ≤ a and θ ≲ 1 is bounded and the
+integrand of eq. (3.75) is bounded in this region, the
+contribution to ⟨N⟩ from ω ≲ a is bounded by a constant
+that is independent of T. We may therefore discard this
+contribution. In the region ω > a, the third Θ-function
+in eq. (3.74) is redundant, and the integration region is
+a ≤ω≤ aeaT/2/2
+(3.76)
+2e−aT/2 ≤θ≤ a
+ω .
+(3.77)
+For aT ≫ 1, we will make negligible error by replacing
+the lower limit of θ by 0. We thereby obtain
+⟨N⟩ ∼
+�qd
+a
+�2 a exp(aT/2)/2
+�
+a
+dω
+a/ω
+�
+0
+dθ θ3ω3
+�
+ln ωθ
+a
+�2
+.
+(3.78)
+Making the change of variables from θ to
+x = ω
+a θ
+(3.79)
+we find that the θ-integral becomes
+a/ω
+�
+0
+dθ θ3ω3
+�
+ln ωθ
+a
+�2
+= a
+ω a3
+1
+�
+0
+dx x3(ln x)2 ∼ a4
+ω .
+(3.80)
+Thus, we obtain
+⟨N⟩ ∼
+�qd
+a
+�2
+a4
+a exp(aT/2)/2
+�
+a
+dω
+ω
+∼ a2q2d2 ln[exp(aT/2)]
+∼ a3q2d2T.
+(3.81)
+This estimate agrees with eq. (3.23).
+Thus, we have succeeded—with considerable effort!—in
+our goal of deriving the decoherence of Alice’s superpo-
+sition by entirely conventional means. It is notable how
+much simpler the calculation of sec. 3.1 was compared to
+the calculation that we have just completed.
+4.
+COSMOLOGICAL HORIZONS DECOHERE
+QUANTUM SUPERPOSITIONS
+In this section, we apply our analysis to de Sitter space-
+time.
+The de Sitter metric in a static patch is given
+by
+ds2 = −f(r)dt2 + f(r)−1dr2 + r2qABdxAdxB
+(4.1)
+where, in this section, xA are angular coordinates on the
+2-sphere, qAB is the unit round metric on the 2-sphere,
+and
+f(r) = 1 − r2/R2
+H
+(4.2)
+where RH (the “Hubble radius”) is a constant.
+The
+coordinate singularity at r = RH corresponds to the
+“cosmological horizon,” which is a Killing horizon of the
+static Killing field (∂/∂t)a. The relation between “affine
+time,” V , and “Killing time,” v, on the future cosmological
+horizon is
+V = ev/RH.
+(4.3)
+The general analysis of sec. 2 applies to the decoherence
+of a static superposition in de Sitter spacetime. The esti-
+mates of the decoherence due to emission of soft photons
+and gravitons through the cosmological horizon when Al-
+ice keeps the superposition present for a time T can be
+made in exact parallel with the analysis of sec. 3 in the
+Rindler case and [14] in the black hole case. The only
+noteworthy new ingredient in de Sitter spacetime is that
+
+14
+the worldline r = 0 is an orbit of the static Killing field
+that is inertial, i.e., non-accelerating. We now estimate
+the decoherence of a spatial superposition created in Al-
+ice’s lab at r = 0 and thereby show that decoherence will
+occur even though Alice’s lab is not accelerating.
+By Gauss’ law, a point charge placed at r = 0 will give
+rise to a radial electric field EU on the future cosmological
+horizon given by
+EU ∼
+q
+R2
+H
+(4.4)
+where EU = Fabℓanb on the horizon with na = (∂/∂V )a
+tangent to the affinely parametrized null generators of
+the horizon and ℓa = (∂/∂U)a a radial null vector with
+naℓa = −1. The change in the electric field on the horizon
+resulting from a displacement of the charge to r = d ≪
+RH is
+∆EU ∼ qd
+R3
+H
+.
+(4.5)
+By paralleling the steps that led to eq. (3.18) above, we
+find that the change in the tangential components of the
+vector potential at the horizon is
+|∆AA| ≡
+�
+R−2
+H qAB∆AA∆AB
+�1/2 ∼ qd
+R2
+H
+.
+(4.6)
+By paralleling the steps that led to eq. (3.23)—assuming
+that the electromagnetic field is initially in the de Sitter
+invariant vacuum (see footnote 7)—we obtain the estimate
+⟨N⟩ ∼ q2d2
+R3
+H
+T
+(de Sitter, EM) .
+(4.7)
+Thus, restoring constants, the decoherence time due to
+the presence of the cosmological horizon is
+TD ∼ ℏϵ0R3
+H
+q2d2
+(de Sitter, EM) .
+(4.8)
+Since d ≪ RH, the decoherence time will be much larger
+than the Hubble time RH/c unless q is extremely large
+relative to the Planck charge qP ≡ √ϵ0ℏc. Nevertheless,
+we see that decoherence does occur despite the fact that
+Alice’s lab is inertial.
+A similar analysis applies in the gravitational case for
+a spatial superposition of a massive particle in Alice’s lab
+at r = 0. In parallel with the derivation given in sec. 3.1
+above, we find
+⟨N⟩ ∼ m2d4
+R5
+H
+T
+(de Sitter, GR)
+(4.9)
+which leads to a decoherence time
+T GR
+D
+∼
+ℏR5
+H
+Gm2d4
+(de Sitter, GR) .
+(4.10)
+ACKNOWLEDGMENTS
+D.L.D. acknowledges support as a Fannie and John
+Hertz Foundation Fellow holding the Barbara Ann Cana-
+van Fellowship and as an Eckhardt Graduate Scholar
+in the Physical Sciences Division at the University of
+Chicago. This research was supported in part by NSF
+Grant No. 21-05878 to the University of Chicago.
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+Generative Slate Recommendation with Reinforcement
+Learning
+Romain Deffayet
+Naver Labs Europe
+Meylan, France
+University of Amsterdam
+Amsterdam, The Netherlands
+romain.deffayet@naverlabs.com
+Thibaut Thonet
+Naver Labs Europe
+Meylan, France
+thibaut.thonet@naverlabs.com
+Jean-Michel Renders
+Naver Labs Europe
+Meylan, France
+jean-michel.renders@naverlabs.com
+Maarten de Rijke
+University of Amsterdam
+Amsterdam, The Netherlands
+m.derijke@uva.nl
+ABSTRACT
+Recent research has employed reinforcement learning (RL) algo-
+rithms to optimize long-term user engagement in recommender
+systems, thereby avoiding common pitfalls such as user boredom
+and filter bubbles. They capture the sequential and interactive na-
+ture of recommendations, and thus offer a principled way to deal
+with long-term rewards and avoid myopic behaviors. However, RL
+approaches are intractable in the slate recommendation scenario
+– where a list of items is recommended at each interaction turn –
+due to the combinatorial action space. In that setting, an action
+corresponds to a slate that may contain any combination of items.
+While previous work has proposed well-chosen decompositions
+of actions so as to ensure tractability, these rely on restrictive and
+sometimes unrealistic assumptions. Instead, in this work we pro-
+pose to encode slates in a continuous, low-dimensional latent space
+learned by a variational auto-encoder. Then, the RL agent selects
+continuous actions in this latent space, which are ultimately de-
+coded into the corresponding slates. By doing so, we are able to
+(i) relax assumptions required by previous work, and (ii) improve
+the quality of the action selection by modeling full slates instead
+of independent items, in particular by enabling diversity. Our ex-
+periments performed on a wide array of simulated environments
+confirm the effectiveness of our generative modeling of slates over
+baselines in practical scenarios where the restrictive assumptions
+underlying the baselines are lifted. Our findings suggest that repre-
+sentation learning using generative models is a promising direction
+towards generalizable RL-based slate recommendation.
+CCS CONCEPTS
+• Information systems → Recommender systems.
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than the
+author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or
+republish, to post on servers or to redistribute to lists, requires prior specific permission
+and/or a fee. Request permissions from permissions@acm.org.
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+© 2023 Copyright held by the owner/author(s). Publication rights licensed to ACM.
+ACM ISBN 978-1-4503-9407-9/23/02...$15.00
+https://doi.org/10.1145/3539597.3570412
+KEYWORDS
+Slate recommendation, Reinforcement learning, Variational auto-
+encoder
+ACM Reference Format:
+Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, and Maarten de
+Rijke. 2023. Generative Slate Recommendation with Reinforcement Learn-
+ing. In Proceedings of the Sixteenth ACM International Conference on Web
+Search and Data Mining (WSDM ’23), February 27-March 3, 2023, Singa-
+pore, Singapore. ACM, New York, NY, USA, 9 pages. https://doi.org/10.1145/
+3539597.3570412
+1
+INTRODUCTION
+Ubiquitous in online services, recommender systems (RSs) play a
+key role personalization by catering to users’ identified tastes. Ide-
+ally, they also diversify their offerings and help users discover new
+interests [19]. In the latter case, RSs take on an active role, which
+means that recommendations influence future user behavior, and
+therefore their effects on users must be explicitly controlled. Such
+effects can be detrimental: users may get bored if too many simi-
+lar recommendations are made, and it has been well-documented
+that users can end up in so-called filter bubbles or echo chambers
+[4, 13, 28]. From the perspective of the online platform or the con-
+tent provider, user boredom leads to poor retention and conversion
+rates [17], while filter bubbles raise fairness and ethical issues for
+which providers can be held accountable [26]. Conversely, RSs can
+also positively impact users, for example, when users get interested
+in new, unexpected topics or when the RS offers a fair represen-
+tation of available options [1]. It is natural, therefore, to balance
+exploitation (i.e., sticking to the known interests of the user) and
+exploration (i.e., further probing the user’s interests) so as to avoid
+always recommending similar items, and encourage recommenda-
+tions that boost future engagement. The reinforcement learning
+(RL) literature has proposed models and algorithms that aim to
+optimize long-term metrics by acknowledging the causal effect of
+recommendations on users [8, 36].
+In this work we consider the common scenario of slate recom-
+mendation [8, 18, 31], which comes with specific challenges. At each
+interaction turn, a slate recommender system recommends a list of
+items from the collection, and the user interacts with zero, one or
+several of those items. As a consequence, users may not examine
+arXiv:2301.08632v1  [cs.IR]  20 Jan 2023
+
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke
+all the recommended items, which leads to biases in the observed
+interactions along with a complex interplay between items in the
+same slate [27]. More importantly, the size of the action space, i.e.,
+the number of possible slates, prohibits the use of off-the-shelf RL
+approaches [12]. Indeed, as slate recommendation is a combinato-
+rial problem, the evaluation of all actions by the RL agent through
+trial and error is simply intractable: even with as few as 1, 000
+items in the collection, the number of possible slates of size 10 is
+approximately 9.6 × 1029. We propose to tackle this problem in
+the context of a practical scenario, (S), which fits the second-stage
+ranking phase [11] of many content recommendation platforms:
+(S) The collection contains around a thousand items, and at each
+turn of interaction the proposed model must select and rank
+10 items to be presented to the user.
+All our tractability and feasibility statements in this paper must
+therefore be understood through the lens of this scenario (S).
+To reduce the prohibitively large size of the combinatorial action
+space, previous studies have proposed to decompose slates in a
+tractable manner [8, 18, 31] – but at the cost of restrictive assump-
+tions, e.g., concerning mutual independence of items in the slate,
+knowledge of the user click model, availability of high-quality item
+embeddings, or that at most one item per slate is clicked.
+In contrast, in this work we propose to first learn a continuous,
+low-dimensional latent representation of actions (i.e., slates), and
+then let the agent take actions within this latent space during its
+training phase. In practice, we obtain the latent representations
+by introducing a generative modeling of slates (GeMS) based on a
+variational auto-encoder (VAE) pre-trained on a dataset of observed
+slates and clicks, collected from a previous version of the recom-
+mender system. Such a dataset is usually available in industrial
+recommendation settings. Therefore, we do not rely on restrictive
+assumptions, and the fact that we represent full slates enables the
+agent to improve the quality of its recommendations, instead of
+using individual item representations.
+Our contributions can be summarized as follows:
+• We propose GeMS, a novel way to represent actions in RL for slate
+recommendation, by pre-training a VAE on slates and associated
+clicks. Unlike previous methods, GeMS is free of overly restrictive
+assumptions and only requires logged interaction data.
+• We provide a unified terminology to classify existing slate recom-
+mendation approaches based on their underlying assumptions.
+• We show on a wide array of simulated environments that previ-
+ous methods underperform when their underlying assumptions
+are lifted (i.e., in practical settings), while GeMS allows us to re-
+cover highly rewarding policies without restrictive assumptions.
+• To support the reproducibility of this work, we publicly release
+the code for our approach, baselines and simulator.1
+2
+RELATED WORK
+Long-term user engagement. Several studies have documented
+the misalignment between short-term benefits and long-term user
+engagement [1, 17], as well as the tendency of traditional recom-
+mender systems to be detrimental to long-term outcomes [29]. Such
+myopic behavior is known to cause boredom and decrease user re-
+tention [1], which is prejudicial for both users and content providers.
+1https://github.com/naver/gems.
+This behavior also raises concerns such as the rich-get-richer issue
+[8] and feeding close-mindedness [29]. Some previous studies tried
+to counter this effect by explicitly maximizing diversity [33] or
+by finding metrics correlated with long-term outcomes [2, 7]. In
+contrast, in our work we directly optimize long-term metrics by
+using reinforcement learning algorithms [8, 16, 36].
+Reinforcement learning for slate recommendation. The prob-
+lem of slate recommendation with reinforcement learning (RL) has
+been tackled in several previous studies, although the settings in
+which solutions were tested vary and are sometimes not applicable
+to our scenario (S). Chen et al. [8] and Bai et al. [3] assume a simple
+user click model and independence of items within a slate in order
+to reduce the problem to choosing individual items, which they
+solve with the REINFORCE algorithm on a SoftMax policy. Ie et al.
+[18] assume knowledge of the user’s click model and item relevance,
+which allows them to perform combinatorial optimization for the
+computation of Q-values. Sunehag et al. [31] take a continuous
+action in the product space of item embeddings, i.e., one embed-
+ding per slot in the slate, and pre-select nearest-neighbor items
+for full-slate Q-function evaluation. Chen et al. [9] use properties
+of the optimal Q-function to propose an elegant decomposition
+of it and generate optimal slates autoregressively. We detail the
+assumptions made by each of these approaches in Section 4, but
+we had to discard [9] due to its prohibitively heavy computation: it
+requires a number of neural network forward passes proportional
+to the slate size times the number of items in the collection (i.e.,
+10,000 passes in scenario (S)), for each training or inference step.
+Our proposed approach differs from previous work because we
+do not manually decompose the slates using tractable heuristics
+based on restrictive assumptions, but instead approximate the slate
+generation process with a deep generative model. Our proposed
+framework only has a single requirement, viz. the availability of
+logged data with slates and associated clicks, as we will detail in
+Section 4. The latter assumption is by no means restrictive as such
+logged data is readily available in common industrial recommenda-
+tion settings.
+Latent action representations. While learning a latent repre-
+sentation of states is very common in the RL literature [14, 30],
+few studies have tackled the problem of latent action representa-
+tion. Chandak et al. [6] train an action generation function in a
+supervised manner, by learning to predict the action taken from
+a pair of successive states. This is not directly applicable in our
+case, because the true user state is not observable and successive
+observations are simply clicks that appear to be too weak of a signal
+to infer the slates leading to these clicks. Botteghi et al. [5] learn a
+state-action world model and jointly train latent state and action
+representations in a model-based fashion.
+Learning a world model in our setting essentially amounts to the
+latent modeling of slates and clicks (similar to our approach), while
+also conditioning on an internal hidden state.2 The work by Zhou
+et al. [35] is perhaps the closest work to ours, as it uses a variational
+auto-encoder (VAE) to embed actions into a controllable latent space
+before training an RL agent. However, it does not consider slates
+but only simple, atomic actions. In contrast, Jiang et al. [20], Liu
+2We tried a similar method in pilot experiments, but the additional conditioning only
+deteriorated the results, so we only present the condition-free method in this paper.
+
+Generative Slate Recommendation with Reinforcement Learning
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Figure 1: Our proposed framework for slate recommendation with reinforcement learning. We first pretrain our GeMS model on previously
+collected logged data composed of slates and associated clicks (left), then we use the frozen decoder of GeMS to decode the RL agent’s low-
+dimensional proto-action vector into a slate (right).
+et al. [25] train VAEs to represent slates and their associated clicks,
+but they do not investigate training an RL agent from the learned
+latent representation.
+To the best of our knowledge, we are the first to learn a latent
+representation of slates for RL-based recommendation.
+3
+METHOD
+3.1
+Notations and problem definition
+We consider a slate recommendation scenario in which a user inter-
+acts with a recommender system (RS) throughout an episode of 𝑇
+turns. At every turn 𝑡 ∈ {1, . . . ,𝑇 }, the system recommends a slate
+𝑎𝑡 = (𝑖1
+𝑡 , . . . ,𝑖𝑘
+𝑡 ) where (𝑖 𝑗
+𝑡 )1⩽𝑗⩽𝑘 are items from the collection I
+and 𝑘 is the size of the slate set by the RS designer. The user can
+click on zero, one or several items in the slate and the resulting
+click vector 𝑐𝑡 = (𝑐1
+𝑡 , . . . ,𝑐𝑘
+𝑡 ),𝑐 𝑗
+𝑡 ∈ {0, 1} is returned to the RS.
+The problem of maximizing the cumulative number of clicks
+over an episode can be modeled as a partially observable Markov
+decision process (POMDP) M𝑃 = (S, O, A, 𝑅,𝑇, Ω) defined by:
+• A set of states S, which represent the unobservable state of the
+user’s mind;
+• A set of observations O accessible to the system. Here, obser-
+vations are clicks from the previous interaction (𝑜𝑡 = 𝑐𝑡−1) and
+therefore lie in the space of binary vectors of size 𝑘: O = {0, 1}𝑘;
+• A set of actions A, which is the set of all possible slates composed
+of items from the collection, i.e., |A| =
+|I |!
+( |I |−𝑘)!;
+• A reward function 𝑅 : S × A → R, which we set to 𝑅(𝑠𝑡,𝑎𝑡) =
+𝑟𝑡 = �𝑘
+𝑗=1 𝑐 𝑗
+𝑡 in order to reflect our long-term objective of maxi-
+mizing the cumulative number of clicks; and
+• A set of unknown transition and observation probabilities, re-
+spectively 𝑇 : S × A × S → [0, 1] and Ω : S × A × O → [0, 1],
+as well as a distribution over initial states 𝑆1 : S → [0, 1].
+Due to the unobserved nature of the true user state in the POMDP, it
+is common to train agents by relying on a proxy of the state inferred
+from available observations. The function that provides such proxy
+is traditionally referred to as the belief encoder [21]. We also define
+the concepts of a policy 𝜋 : S × A → [0, 1] and trajectory 𝜏 =
+(𝑜𝑡,𝑎𝑡,𝑟𝑡)1⩽𝑡⩽𝑇 . In the remainder, we write 𝜏 ∼ 𝜋 to signify that
+we obtain a trajectory by first sampling an initial state 𝑠1 from 𝑆1
+and then recursively sampling actions𝑇 −1 times from the policy 𝜋.
+The goal can now be formulated as finding an optimal policy, i.e., a
+policy maximizing the expected return 𝜋∗ ∈ arg max𝜋 E𝜏∼𝜋 [R(𝜏)]
+with R(𝜏) = �𝑇
+𝑡=1 𝑟𝑡. Finally, given a state 𝑠 and action 𝑎, we define
+the Q-function 𝑄𝜋 (𝑠,𝑎) = E𝜏∼𝜋,𝑠1=𝑠,𝑎1=𝑎 [R(𝜏)] and V-function
+𝑉 𝜋 (𝑠) = E𝑎∼𝜋 (𝑠) [𝑄𝜋 (𝑠,𝑎)].
+3.2
+Overview of the framework
+In our proposed framework, the interactions with the environment,
+i.e., the user, can be described by the following repeated steps:
+(1) The belief encoder summarizes the history of interactions with
+the user into a state vector;
+(2) The agent selects a proto-action based on this state; and
+(3) The ranker (here resulting from a VAE model) decodes this
+proto-action into a slate that is served to the user.
+In the remainder of this section, we first detail our proposed gener-
+ative modeling of slates (GeMS). GeMS is a deep generative model
+that learns a low-dimensional latent space for slates and associated
+clicks – thus constituting a convenient proto-action space for the RL
+agent and allowing for tractable RL without resorting to restrictive
+assumptions as in prior work [3, 8, 18, 31]. Then we describe how
+GeMS is integrated as a ranker in our RL framework and we briefly
+discuss the remaining RL components. This two-step process is
+depicted in Figure 1.
+3.3
+Generative Modeling of Slates (GeMS)
+In order to instantiate our GeMS model, we propose to train a vari-
+ational auto-encoder (VAE) on a precollected dataset D of logged
+interactions, as illustrated in Figure 1 (left). A VAE aims to learn
+a joint distribution over data samples (i.e., slates and clicks de-
+noted as 𝑎 and 𝑐, respectively) and latent encodings (i.e., proto-
+actions denoted as 𝑧) [22]. To do so, a parameterized distribution
+𝑝𝜃 (𝑎,𝑐,𝑧) is trained to maximize the marginal likelihood of the data
+𝑝𝜃 (𝑎,𝑐) =
+∫
+𝑧 𝑝𝜃 (𝑎,𝑐,𝑧)𝑑𝑧. In practice, due to the intractability of
+this integral, a parameterized distribution 𝑞𝜙 (𝑧|𝑎,𝑐) is introduced
+as a variational approximation of the true posterior 𝑝𝜃 (𝑧|𝑎,𝑐) and
+the VAE is trained by maximizing the evidence lower bound (ELBO):
+LELBO
+𝜃,𝜙
+=E𝑎,𝑐∼D
+�
+E𝑧∼𝑞𝜙 (·|𝑎,𝑐) [log 𝑝𝜃 (𝑎,𝑐|𝑧)]−KL
+�
+𝑞𝜙 (𝑧|𝑎,𝑐)∥𝑝(𝑧)
+��
+,
+where 𝑝(𝑧) is the prior distribution over the latent space, KL is the
+Kullback-Leibler divergence [24], and 𝑧 is a sample from a Gaussian
+
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke
+distribution obtained using the reparameterization trick [22]. The
+distributions 𝑞𝜙 (𝑧|𝑎,𝑐) and 𝑝𝜃 (𝑎,𝑐|𝑧) are usually referred to as the
+encoder and the decoder, respectively.
+The downstream performance of the RL agent we wish to ulti-
+mately learn clearly depends on the upstream ability of the VAE
+to properly reconstruct slates. However, as Liu et al. [25] observe,
+an accurate reconstruction of slates may limit the agent’s capacity
+to satisfy the user’s interests. Indeed, finding high-performance
+continuous control policies requires smoothness and structure in
+the latent space, which may be lacking if too much emphasis is
+given to the reconstruction objective in comparison to the prior
+matching objective enforced by the KL-divergence. Therefore, it
+is necessary to balance reconstruction and controllability, which
+is done by introducing an hyperparameter 𝛽 as weight for the KL
+term in Eq. ??. Moreover, in order to promote additional structure
+in the latent space, we add a click reconstruction term in the loss:
+slates with similar short-term outcomes (i.e., clicks) are grouped
+together during pre-training. Yet, we may want to avoid biasing
+the learned representations towards click reconstruction too much,
+as it may come at the cost of quality of the slate reconstruction.
+Therefore, we introduce a hyperparameter 𝜆 to adjust this second
+trade-off. We show the empirical impact of 𝛽 and 𝜆 in Section 6.3.
+In our implementation, the prior 𝑝(𝑧) is set as a standard Gauss-
+ian distribution N (0, I). The encoder 𝑞𝜙 (𝑧|𝑎,𝑐) is a Gaussian dis-
+tribution with diagonal covariance N (𝜇𝜙 (𝑎,𝑐), diag(𝜎2
+𝜙 (𝑎,𝑐))), pa-
+rameterized by a multi-layer perceptron (MLP). This MLP inputs
+the concatenation of learnable item embeddings and associated
+clicks over the whole slate, and outputs (𝜇𝜙 (𝑎,𝑐), log 𝜎𝜙 (𝑎,𝑐)). For
+the decoder 𝑝𝜃 (𝑎,𝑐|𝑧), another MLP takes as input the latent sam-
+ple 𝑧, and outputs the concatenation of reconstructed embeddings
+e𝑗
+𝜃 (𝑧) and click probabilities 𝑝 𝑗,𝑐
+𝜃 (𝑐𝑗 |𝑧) for each slot 𝑗 in the slate.
+We then derive logits for the item probabilities 𝑝 𝑗,𝑎
+𝜃 (𝑎𝑗 |𝑧) by taking
+the dot-product of the reconstructed embedding e𝑗
+𝜃 (𝑧) with the
+embeddings of all items in the collection. For collection items, we
+use the current version of embeddings learned within the encoder,
+but we prevent the gradient from back-propagating to them using
+the stop-gradient operator to avoid potential degenerate solutions.
+In summary, the VAE is pre-trained by maximizing the ELBO on
+the task of reconstructing slates and corresponding clicks, i.e., by
+minimizing LGeMS
+𝜃,𝜙
+= E𝑎,𝑐∼D [LGeMS
+𝜃,𝜙
+(𝑎,𝑐)] with:
+LGeMS
+𝜃,𝜙
+(𝑎,𝑐) =
+slate reconstruction
+������������������������������������������������������
+𝑘
+∑︁
+𝑗=1
+log 𝑝 𝑗,𝑎
+𝜃 (𝑎𝑗 |𝑧𝜙 (𝑎,𝑐)) +
+𝜆
+click reconstruction
+������������������������������������������������������
+𝑘
+∑︁
+𝑗=1
+log 𝑝 𝑗,𝑐
+𝜃 (𝑐𝑗 |𝑧𝜙 (𝑎,𝑐)) +
+𝛽
+KL-divergence
+������������������������������������������������������������������������
+𝑑
+∑︁
+𝑖=1
+�
+𝜎2
+𝜙,𝑖 + 𝜇2
+𝜙,𝑖 − log 𝜎𝜙,𝑖 − 1
+�
+(1)
+where 𝑧𝜙 (𝑎,𝑐) = 𝜇𝜙 (𝑎,𝑐) + diag(𝜎2
+𝜙 (𝑎,𝑐)) · 𝜖, for 𝜖 ∼ N (0, I). Here,
+𝑑 is the dimension of the latent space, and 𝛽 and 𝜆 are hyperparam-
+eters controlling the respective weight of the KL term and the click
+reconstruction term. Note that the KL term takes this simple form
+due to the Gaussian assumption on 𝑞𝜙 (𝑧|𝑎,𝑐) and the N (0, I) prior.
+3.4
+RL agent and belief encoder
+After the pre-training step described in Section 3.3, the parameters
+of GeMS are frozen and we use its decoder as the ranker in our
+RL framework. The RL agent can then be trained to maximize the
+discounted return by taking proto-actions within the VAE’s latent
+space. To generate a slate (𝑖1, . . . ,𝑖𝑘) from the agent’s proto-action
+𝑧, we take for each slot 𝑗 ∈ {1, . . . ,𝑘} the most likely item according
+to the decoder: 𝑖 𝑗 = arg max𝑖 ∈I 𝑝 𝑗,𝑎
+𝜙 (𝑖|𝑧).
+Since our focus within the RL framework is on the choice of the
+ranker, we adopt a standard implementation of the belief encoder
+and the agent: the former is modeled by a gated recurrent unit
+(GRU) [10] taking as input the concatenation of item embeddings
+and respective clicks from each slate, and the latter is a soft actor-
+critic (SAC) [15] algorithm. We chose SAC because it is a well-
+established RL algorithm, known for its strong performance and
+data-efficiency in continuous control. Additionally, SAC adds an
+entropy term incentivizing exploration which we have noticed
+during our experiments to be important to attain high performance
+in highly stochastic recommendation environments.
+4
+BASELINES AND THEIR ASSUMPTIONS
+We evaluate our proposed method against four main baselines
+derived from prior work. In this section, we describe these baselines
+as well the assumptions on user behavior that they formulate in
+order to make the combinatorial problem of slate recommendation
+tractable. By doing so, we are able to compare the assumptions
+made by these baselines and highlight the generality of our method
+in Table 1. Note that we only report from previous studies the
+mechanism used for slate generation, which is the topic of this
+study, and ignore other design choices.
+SoftMax. In [3, 8], the authors reduce the combinatorial problem
+of slate optimization to the simpler problem of item optimization:
+the policy network output is a softmax layer over all items in the
+collection, and items are sampled with replacement to form slates.
+Doing so requires the mild assumption that the Q-value of the slate
+can be linearly decomposed into item-specific Q-values (DQ). But
+more importantly, it also requires two strong assumptions, namely
+users can click on at most one item per slate (1CL) and the returns
+of items in the same slate are mutually independent (MI). Together,
+these assumptions are restrictive, because their conjunction means
+that the click probability of an item in the slate does not depend
+on the item itself. Indeed, having dependent click probabilities
+(to enforce the single click) and independent items in the slate is
+compatible only if click probabilities do not depend on items.
+SlateQ. Ie et al. [18] propose a model-based approach in which
+the click behavior of the user is given, and Q-learning [34] is used
+to plan and approximate users’ dynamic preferences. On top of
+the earlier DQ and 1CL, it requires access to the true relevance and
+click model (CM), which is an unfair advantage compared to other
+methods. For computational efficiency reasons, we adopt the faster
+variant referred to as QL-TT-TS in the original paper.
+TopK. Even though, to the best of our knowledge, no work has
+proposed this approach, we include it in our set of baselines as
+
+Generative Slate Recommendation with Reinforcement Learning
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Table 1: Comparison of assumptions made by prior work. Our
+method only requires access to logged interaction data.
+1CL
+DQ
+MI
+CM
+SP
+EIB
+LD
+SoftMax [3, 8]
+✓
+✓
+✓
+✗
+✗
+✗
+✗
+SlateQ [18]
+✓
+✓
+✗
+✓
+✗
+✗
+✗
+WkNN [31]
+✗
+✓
+✗
+✗
+✓
+✓
+✓
+TopK
+✗
+✗
+✗
+✗
+✓
+✗
+✓
+GeMS (Ours)
+✗
+✗
+✗
+✗
+✗
+✗
+✓
+it is a natural way to deal with slate recommendation. The agent
+takes continuous actions in the space of item embeddings, and we
+generate slates by taking the 𝑘 items from the collection with the
+closest embeddings to the action, according to a similarity metric
+(the dot-product in practice). This method therefore assumes the
+availability of logged data of past interactions (LD), in order to
+pre-train item embeddings. In our experiments, we evaluate two
+variants of this baseline: TopK (MF), where item embeddings are
+learned by matrix factorization [23], and TopK (ideal), which uses
+ideal item embeddings, i.e., the embeddings used internally by the
+simulator (see Section 5.1). The latter version clearly has an unfair
+advantage. Also, because ranking items this way assumes that the
+most rewarding items should appear on top, it makes the sequential
+presentation (SP) assumption from [31] that the true click model
+is top-down and fading, i.e., if 𝑐(𝑖) indicates that item 𝑖 has been
+clicked and 𝑙 ⩽ 𝑘 is the position of 𝑖 in slate 𝑎, then 𝑃(𝑐(𝑖)|𝑠,𝑎) =
+𝑃(𝑐(𝑖)|𝑠,𝑎⩽𝑙) ⩽ 𝑃(𝑐(𝑖)|𝑠, ˜𝑎⩽𝑙−1), where 𝑎⩽𝑙 = (𝑖1, . . . ,𝑖𝑙−1,𝑖) and
+˜𝑎⩽𝑙−1 = (𝑖1, . . . ,𝑖𝑙−2,𝑖).
+WkNN. In [31], the authors propose a finer-grained and potentially
+more capable variant of TopK referred to as Wolpertinger [12]: the
+agent takes actions in the product-space of item embeddings over
+slate slots, i.e., continuous actions of dimension 𝑘 ×𝑑, where 𝑑 is the
+dimension of item embeddings. Then, for each slot in the slate, 𝑝
+candidate items are selected by Euclidean distance with embeddings
+of items from the collection, and every candidate item’s contribution
+to the Q-value is evaluated in a greedy fashion. Besides LD and DQ,
+WkNN requires two strong assumptions to ensure submodularity
+of the Q-function: sequential presentation SP and execution is best
+(EIB), i.e., recommendations that are risky on the short term are
+never worth it. Formally, this translates as: P(𝑅(𝑠, 𝜋1(𝑠)) = 0) ⩾
+P(𝑅(𝑠, 𝜋2(𝑠)) = 0) ⇒ 𝑉 𝜋1 (𝑠) ⩽ 𝑉 𝜋2 (𝑠) for any policies 𝜋1, 𝜋2.
+Note that it partly defeats the purpose of long-term optimization.
+In Table 1, we summarize the assumptions made by each baseline.
+In comparison to prior work, our proposed framework has a single
+assumption: the availability of logged data with slates and asso-
+ciated clicks (LD), as Table 1 indicates. This assumption is by no
+means restrictive as such logged data is readily available in common
+industrial recommendation settings.
+On top of these baselines, we also include a random policy and
+a short-term oracle as reference points. The short-term oracle
+has access to the true user and item embeddings, enabling it to
+select the items with the highest relevance probability in each slate.
+Therefore, at each turn of interaction, it gives an upper bound on
+the immediate reward but it is unable to cope with boredom and
+influence phenomena.
+5
+EXPERIMENTAL SETUP
+5.1
+Simulator
+We design a simulator that allows us to observe the effect of lifting
+the assumptions required by the baselines, and we experiment with
+several simulator variants to ensure generalizability. We summarize
+our main design choices below and refer the reader to our code
+available online3 for a more detailed description.
+Item and user embeddings. Following scenario (S), our simula-
+tor includes 1, 000 items. We consider a cold-start situation where
+users are generated on-the-fly for each new trajectory. Items and
+users are randomly assigned embeddings of size 20, corresponding
+to ten 2-dimensional topics: e = (e1, . . . , e10). Each 2-dimensional
+vector e𝑡 is meant to capture the existence of subtopics within
+topic 𝑡. The embedding of a user or item 𝑥 is generated using the
+following process: (i) sample topic propensities 𝑤𝑡𝑥 ∼ U(0, 1) and
+normalize such that �
+𝑡 𝑤𝑡𝑥 = 1; (ii) sample topic-specific compo-
+nents 𝜖𝑡𝑥 ∼ N (0, 0.4 · I2) and rescale as e𝑡𝑥 = 𝑤𝑡𝑥 · min(|𝜖𝑡𝑥 |, 1));
+and (iii) normalize the embedding e𝑥 = (e1𝑥, . . . , e10
+𝑥 ) such that
+∥e𝑥 ∥ = 1. Each item is associated to a main topic, defined as
+𝑡(𝑖) = arg max1⩽𝑡⩽10 ∥e𝑡
+𝑖 ∥.
+To accomodate different types of content and platforms, we
+derive two variants of item embeddings in the simulator: one with
+embeddings obtained as described above, and one with embeddings
+for which we square and re-normalize each component. In Section 6,
+we highlight this difference in peakedness by referring to the former
+as diffuse embeddings and the latter as focused embeddings.
+Relevance computation. The relevance probability of item 𝑖 for
+user 𝑢 is a monotonically increasing function of the dot-product
+between their respective embeddings: rel(𝑖,𝑢) = 𝜎(e𝑖𝑇 e𝑢), where
+𝜎 is a sigmoid function.
+Boredom and influence effects. User embeddings can be af-
+fected by two mechanisms: boredom and influence. Each item 𝑖
+clicked by user 𝑢 influences the user embedding in the next interac-
+tion turn as: e𝑢 ← 𝜔e𝑢 +(1−𝜔)e𝑖, where we set 𝜔 = 0.9 in practice.
+Additionally, if in the last 10 items clicked by user 𝑢 five have the
+same main topic 𝑡𝑏, then 𝑢 gets bored with this topic, meaning we
+put e𝑡𝑏
+𝑢 = 0 for 5 turns. These mechanisms have been defined to
+penalize myopic behavior and encourage long-term strategies.
+Click model. Users click on recommended items according to a
+position-based model, i.e., the click probability is the product of
+item-specific attractiveness and rank-specific examination probabil-
+ities: P(𝑐|𝑖,𝑟) = 𝐴𝑖 × 𝐸𝑟. Specifically, we define for an item located
+at rank 𝑟: 𝐸𝑟 = 𝜈𝜀𝑟 + (1 − 𝜈)𝜀𝑘+1−𝑟 with 𝜀 = 0.85. It is a mixture of
+the terms 𝜀𝑟 and 𝜀𝑘+1−𝑟, which respectively capture the top-down
+and bottom-up browsing behaviors. We use two variants of this
+click model in our experiments: TopDown with 𝜈 = 1.0 and Mixed
+with 𝜈 = 0.5. The attractiveness of an item is set to its relevance
+in TopDown and Mixed. In addition, we consider a third variant
+DivPen which also penalizes slates that lack diversity: 𝐴𝑖 is down-
+weighted by a factor of 3 if more than 4 items from the slate have
+the same main topic (as in Mixed, we also set 𝜈 = 0.5 for DivPen).
+In summary, our experiments are performed on 6 simulator variants
+defined by the choice of item embedding peakedness (diffuse item
+3https://naver/github/gems
+
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke
+Table 2: Average cumulative number of clicks on the test set for our 6 simulated environments. Bold: best method; underlined: 2nd-best
+method; †: statistically significantly better than all other methods. 95% confidence intervals are given in parentheses. Methods grouped under
+“Disclosed env.” have access to privileged information about the environment and can therefore not be fairly compared with “Undisclosed
+env.” methods.
+Focused item embeddings
+Diffuse item embeddings
+Method
+TopDown
+Mixed
+DivPen
+TopDown
+Mixed
+DivPen
+Disclosed
+env.
+
+
+Short-term oracle
+SAC+TopK (ideal)
+SlateQ
+107.7
+101.6
+85.4
+96.7
+94.6
+78.8
+429.0 (±5.9)
+384.1 (±13.5)
+386.3 (±15.5)
+373.9 (±25.0)
+371.9 (±36.4)
+341.3 (±55.3)
+206.5 (±4.1)
+202.7 (±3.4)
+119.0 (±3.9)
+209.5 (±5.4)
+192.7 (±5.1)
+117.8 (±5.8)
+Undisclosed
+env.
+
+
+Random
+REINFORCE+SoftMax
+SAC+WkNN
+SAC+TopK (MF)
+SAC+GeMS (Ours)
+33.8 (±0.2)
+33.9 (±0.2)
+33.6 (±0.2)
+33.3 (±0.2)
+33.2 (±0.2)
+32.9 (±0.2)
+248.1 (±19.3)
+233.5 (±18.5)
+249.1 (±11.6)
+249.5 (±15.3)
+214.7 (±25.0)
+213.8 (±27.1)
+98.5 (±8.9)
+97.7 (±10.8)
+95.5 (±9.9)
+107.2 (±8.9)
+89.8 (±7.4)
+92.5 (±5.0)
+254.4 (±17.1)
+232.7 (±19.4)
+242.2 (±15.4)
+249.7 (±10.3)
+184.1 (±1.3)
+231.4 (±13.3)
+305.3†(±21.9)
+242.6 (±21.5)
+254.1 (±27.7)
+300.0†(±42.8)
+260.6†(±27.2)
+249.6 (±37.6)
+embeddings or focused item embeddings) and the choice of click
+model (TopDown, Mixed, or DivPen).
+5.2
+Implementation and evaluation details
+Our implementation aims to be as standard as possible, considering
+the literature on RL, in order to ensure reproducibility. All base-
+lines are paired with SAC [15], except SlateQ which is based on
+Q-Learning [34], and SoftMax, which we pair with REINFORCE [32]
+because it requires a discrete action space and a discretized variant
+of SAC led to lower performance in our experiments. We implement
+all agents using two-layer neural networks as function approxima-
+tors, and use target networks for Q-functions in Slate-Q and SAC.
+For hyperparameters common to baselines and our method, we
+first performed a grid search over likely regions of the space on
+baselines, and re-used the selected values for our method. For all
+methods we use the Adam optimizer with learning rates of 0.001
+for Q-networks and 0.003 for policy networks when applicable, as
+well as a discount factor 𝛾 = 0.8 and a polyak averaging parameter
+𝜏 = 0.002. For the hyperparameters specific to our method (𝑑, 𝛽
+and 𝜆), we perform a grid search on the TopDown environment
+with focused item embeddings and select the combination with
+the highest validation return. This combination is then re-used
+on all other environments. The searched ranges were defined as
+𝑑 ∈ {16, 32}, 𝛽 ∈ {0.1, 0.2, 0.5, 1.0, 2.0} and 𝜆 ∈ {0.0, 0.2, 0.5, 1.0}.
+For methods making the (LD) assumption, we generated a dataset
+of 100K user trajectories (with 100 interactions turns each) from an
+𝜖-greedy oracle policy with 𝜖 = 0.5, i.e., each recommended item is
+selected either uniformly randomly or by an oracle, with equal prob-
+abilities. The VAE in GeMS is trained on this dataset for 10 epochs
+with a batch size of 256 and a learning rate of 0.001. For approaches
+requiring pre-trained item embeddings (TopK and WkNN), we learn
+a simple matrix factorization model on the generated dataset by
+considering as positive samples the pairs composed of the user in
+the trajectory and each clicked item in their recommended slates.
+In all of our experiments, we compare average cumulative re-
+wards over 10 seeded runs, corresponding to ten initializations of
+the agent’s parameters. In the case of GeMS, the seed also controls
+the initialization of the VAE model during pre-training. We train
+agents for 100K steps. Each step corresponds to a user trajectory,
+composed of 100 interaction turns (i.e., 100 slates successively pre-
+sented to the user) for a unique user. Every 1, 000 training steps, we
+also evaluate the agents on 200 validation user trajectories. Finally,
+the agents are tested by selecting the checkpoint with the highest
+validation return and applying it on 500 test user trajectories. Con-
+fidence intervals use Student’s 𝑡-distribution, and statistical tests
+are Welch’s 𝑡-test. Both are based on a 95% confidence level.
+6
+RESULTS
+In our experiments, we investigate the following research ques-
+tions: (RQ1) How does our slate recommendation framework based
+on GeMS compare to previous methods when the underlying as-
+sumptions of the latter are lifted? (RQ2) Does the proposed GeMS
+framework effectively balance immediate and future rewards to
+avoid boredom? (RQ3) How do the balancing hyperparameters 𝛽
+and 𝜆 in GeMS impact the downstream RL performance?
+6.1
+Comparison of our method against
+baselines (RQ1)
+In this section, we compare the performance of our method and
+baselines on a wide array of simulated environments, corresponding
+to the six environments described in Section 5.1.
+Overview of the results. Table 2 shows the average test return
+(i.e., cumulated reward or cumulated number of clicks) after train-
+ing on 100K user trajectories. We group methods into two cate-
+gories: Disclosed env., i.e., methods leveraging hidden environment
+information, and Undisclosed env., i.e., methods that consider the
+environment as a black-box and are therefore practically applicable.
+A first observation we can draw, regardless of the specific environ-
+ment used, is that the short-term oracle is easily beaten by most
+approaches. Indeed, the simulator penalizes short-sighted recom-
+mendations that lead to boredom: in these environments, diversity
+is required to reach higher returns. We can also observe the superi-
+ority of SAC+TopK (Ideal). This is not surprising, as this method
+benefits from an unfair advantage – access to true item embed-
+dings – but it suggests that practically applicable methods could be
+augmented with domain knowledge to improve their performance.
+However, despite having access to privileged information, SlateQ’s
+performance is subpar, especially in DivPen environments. Its lower
+
+Generative Slate Recommendation with Reinforcement Learning
+WSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+(a) Short-term oracle.
+(b) SAC+GeMS with 𝛾 = 0.
+(c) SAC+GeMS with 𝛾 = 0.8.
+Figure 2: Distribution of the relevance scores of items recommended by (a) a short-term oracle, (b) SAC+GeMS with 𝛾 = 0 and (c) SAC+GeMS
+with 𝛾 = 0.8. Boredom penalizes item scores and is visualized by orange areas. The myopic approaches (left, center) lead to more boredom
+than the long-term approach (right), and therefore to lower average item scores (solid red lines).
+performance might be explained by its approximate optimization
+strategy and restrictive single-click assumption.
+Overall comparison of methods. The proposed SAC+GeMS com-
+pares favorably to baselines across the range of environments we sim-
+ulate. Out of the 6 tested environments, SAC+GeMS obtained the
+best average results on all of them, among which 3 show a statisti-
+cally significant improvement over all other methods. SAC+WkNN
+performs very poorly: we hypothesize that the approach suffers
+from the curse of dimensionality due to the larger action space
+(200 dimensions in our experiments) and the assumption made
+by the approach that candidate items need to be close to target
+item embeddings according to the Euclidean distance. SAC+TopK
+(MF) is more competitive, but the large difference with SAC+TopK
+(ideal) suggests that TopK is very sensitive to the quality of item
+embeddings. Despite its very restrictive assumptions and lack of the-
+oretical guarantees in our setup, REINFORCE+SoftMax was a very
+competitive baseline overall. However, while its best checkpoint
+had high return, its training was unstable and failed to converge in
+our experiments, which suggests it may be unreliable.
+Comparisons across environments. The TopDown environ-
+ment is the easiest for most methods, regardless of the type of
+item embeddings. This is not surprising as all methods besides
+Random either assume a top-down click model, sample items in
+a top-down fashion or rely on data from a top-down logging pol-
+icy. However, it is worth noting that other factors can dominate
+the performance, such as sub-optimality of item embeddings for
+SAC+TopK (MF). Conversely, DivPen was harder for most methods,
+because it requires a strong additional constraint to obtain high
+returns: intra-slate diversity must be high. SAC+GeMS was also af-
+fected by these dynamics, but remained able to beat other methods
+by generating diverse slates. Finally, the use of diffused item embed-
+dings does not appear to cause lower returns for GeMS, compared
+with focused ones, but is associated with larger confidence intervals
+for SAC+GeMS: indeed, pivot items spanning multiple topics are
+more likely to be attractive, at the expense of more fine-grained
+strategies, making the training process uncertain.
+6.2
+GeMS overcomes boredom to improve its
+return (RQ2)
+In Section 1 we highlighted that long-term optimization with RL
+can penalize myopic behavior such as recommending only highly
+relevant but similar items, which may lead to boredom. In this sec-
+tion, we verify that SAC+GeMS is able to adapt its slate selection
+to cope with boredom. We recall that in our simulated environ-
+ments (detailed in Section 5.1), users get bored of a particular topic
+whenever 5 of their latest 10 clicks were on items from that topic.
+When a topic is saturated, its corresponding dimensions in the user
+embedding are set to 0, which has the effect of diminishing the
+attractiveness of future items presented to the user. It is therefore
+necessary to avoid boredom in order to reach higher returns, even
+if it comes at the cost of lower immediate rewards.
+In this section, we compare three approaches on the TopDown
+environment with focused item embeddings: (i) the short-term ora-
+cle (STO) always maximizing the immediate reward, (ii) SAC+GeMS
+with 𝛾 = 0.8 (i.e., our proposed method) where 𝛾 is the discount
+factor of the RL algorithm, and (iii) SAC+GeMS with 𝛾 = 0 which
+does not explicitly include future rewards in its policy gradient. In
+this environment, SAC+GeMS𝛾=0.8 achieves an average test return
+of 305.3, while SAC+GeMS𝛾=0 reaches 194.3, and STO only ob-
+tains 107.7. These results suggest that long-term optimization is
+indeed required to reach higher returns. It may seem surprising
+that SAC+GeMS𝛾=0 gets better returns than STO, but its training
+objective incentivizes average immediate rewards, which implicitly
+encourages it to avoid low future rewards. However, adopting an
+explicit mechanism to account for its causal effect on the user (i.e.,
+setting 𝛾 = 0.8) allows SAC+GeMS to improve its decision-making.
+In Figure 2, we plot the distribution of item scores (i.e., the dot-
+product between internal user and item embeddings as defined in
+Section 5.1) for the items recommended in slates by each of the
+three methods, with the same seed for all three plots. The dashed
+vertical line shows the score threshold of 0.28 needed to reach a
+relevance probability of 0.5. Therefore, items on the left of this
+line have a lower click probability while items on the right have a
+higher click probability. The color indicates how many topics were
+saturated when the agent recommended that particular item whose
+score is plotted: one can see that when the user is bored of at least
+one topic, items become less attractive as scores are reduced.
+When no topic is saturated (i.e., yellow distribution), STO rec-
+
+average score
+ithreshold
+12
+1
+Number of
+10
+saturated topics
+0
+1
+8
+2
+1
+PDF
+6
+4
+2
+00
+0.1
+0.2
+0.3
+0.4
+0.5
+Scoreaverage score
+ithreshold
+12
+Number of
+saturated topics
+10
+0
+1
+2
+8
+PDF
+6
+4
+2
+00
+0.1
+0.2
+0.3
+0.4
+0.5
+Scorethreshold .
+ average score
+18
+Number of
+16
+saturated topics
+14
+0
+1
+12
+2
+PDF
+10
+8
+6
+4
+2
+00
+0.1
+0.2
+0.3
+0.4
+0.5
+ScoreWSDM ’23, February 27-March 3, 2023, Singapore, Singapore
+Romain Deffayet, Thibaut Thonet, Jean-Michel Renders, & Maarten de Rijke
+(a) Impact of 𝛽 for 𝜆 = 0.5.
+(b) Impact of 𝜆 for 𝛽 = 1.0.
+Figure 3: Average cumulative number of clicks on the validation set obtained by SAC+GeMS with its best validation checkpoint, for different
+values of 𝛽 and 𝜆 (defined in Section 3.3). We also display 95% confidence intervals.
+ommends items with excellent scores (above the threshold and
+up to 0.45): as a consequence, STO gets high immediate rewards.
+However, by doing so it incurs a lot of boredom (large orange
+areas). Overall, it leads to lower expected scores (solid red line)
+and therefore fewer clicks. Conversely, SAC+GeMS𝛾=0.8 sacrifices
+some immediate reward (yellow distribution shifted to the left) but
+causes very little boredom (small orange area). Overall, by trading
+off relevance and diversity, SAC+GeMS𝛾=0.8 yields good immediate
+rewards while limiting boredom. It therefore gets higher average
+scores. SAC+GeMS𝛾=0 exhibits an intermediate behavior due to its
+limited capabilities: it recommends items of varying relevance, yet
+leads to substantial boredom (larger orange area than for 𝛾 = 0.8).
+6.3
+Balancing hyperparameters 𝛽 and 𝜆 (RQ3)
+In Section 3.3, we suggested that the choice of 𝛽 and 𝜆 leads to trade-
+offs that may impact the downstream performance of SAC+GeMS.
+As a reminder, 𝛽 adjusts the importance of accurate reconstruction
+versus smoothness and structure in the latent space (i.e., controlla-
+bility), while 𝜆 weights the click reconstruction with respect to the
+slate reconstruction. Next, we verify our intuition on the importance
+of these trade-offs by reporting (in Figure 3) the best validation
+return obtained for different values of said hyperparameters, on
+the TopDown environment with focused item embeddings.
+Figure 3a suggests that, indeed, there exists a “sweet spot” in the
+selection of 𝛽. It confirms the intuition described in Section 3.3 and
+the observation of Liu et al. [25]: 𝛽 must be appropriately balanced
+in order to ensure high performance on the downstream RL task.
+Specifically, we found that choosing 𝛽 = 1.0 leads to the highest
+return overall, regardless of whether a latent dimension of 16 or
+32 is used.
+The impact on the downstream performance of the trade-off
+between slate and click reconstruction (Figure 3b) is less prominent
+but can still be observed. It justifies our choice to add the click
+reconstruction term in the loss (Eq. 1), even though clicks output by
+GeMS’ decoder are not used during RL training. This also confirms
+the importance of introducing and adjusting the hyperparameter 𝜆:
+modeling clicks jointly with slates improves the final performance of
+SAC+GeMS, but properly weighting the click reconstruction objective
+with respect to the slate reconstruction objective is necessary.
+7
+CONCLUSION
+We have presented GeMS, a slate representation learning method
+based on variational auto-encoders for slate recommendation with
+reinforcement learning. This method has the notable advantage
+of being flexible, allowing full-slate modeling and lightweight as-
+sumptions, in contrast with existing approaches.
+Findings and broader impact. Our experiments across a wide
+array of environments demonstrate that GeMS compares favor-
+ably against existing slate representation methods in practical set-
+tings. Moreover, our empirical analysis highlights that it effectively
+balances immediate and future rewards, and that the trade-offs
+imposed by 𝛽 and 𝜆 significantly impact the RL downstream perfor-
+mance, indicating that properly balancing these hyperparameters is
+critical. Our work suggests that generative models are a promising
+direction for representing rich actions such as slates.
+Limitations. Our simulated experiments demonstrate the effec-
+tiveness of GeMS for representing slates in an RL framework. How-
+ever, it is well-known that online training of RL agents is too expen-
+sive and risky, and that in practice agents must be trained offline, i.e.,
+directly from logged data [8]. We did not address here the specific
+challenges of offline RL, as we wished to isolate the contribution of
+the slate representation to downstream performance.
+Future work. In future work, we will investigate how generative
+models can be leveraged in the offline setting, in different scenarios,
+or with even richer actions. We also plan to look into improvements
+of the architectures used for structured action representations, for
+example by using domain knowledge and user models.
+ACKNOWLEDGMENTS
+This research was (partially) funded by the Hybrid Intelligence
+Center, a 10-year program funded by the Dutch Ministry of Educa-
+tion, Culture and Science through the Netherlands Organisation for
+Scientific Research, https://hybrid-intelligence-centre.nl. All con-
+tent represents the opinion of the authors, which is not necessarily
+shared or endorsed by their respective employers and/or sponsors.
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+

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+Multi-Scanner Canine Cutaneous Squamous Cell
+Carcinoma Histopathology Dataset
+Frauke Wilm1,2, Marco Fragoso3, Christof A. Bertram4, Nikolas Stathonikos5,
+Mathias Öttl1, Jingna Qiu2, Robert Klopfleisch3, Andreas Maier1,
+Katharina Breininger2,†, Marc Aubreville6,†
+1Pattern Recognition Lab, Friedrich-Alexander-Universität Erlangen-Nünberg, Germany
+2Department AIBE, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
+3Institute of Veterinary Pathology, Freie Universität Berlin, Germany
+4Institute of Pathology, University of Veterinary Medicine, Vienna, Austria
+5Pathology Department, University Medical Centre Utrecht, The Netherlands
+6Technische Hochschule Ingolstadt, Ingolstadt, Germany
+†shared senior authors
+frauke.wilm@fau.de
+Abstract. In histopathology, scanner-induced domain shifts are known to impede
+the performance of trained neural networks when tested on unseen data. Multi-
+domain pre-training or dedicated domain-generalization techniques can help to
+develop domain-agnostic algorithms. For this, multi-scanner datasets with a high
+variety of slide scanning systems are highly desirable. We present a publicly
+available multi-scanner dataset of canine cutaneous squamous cell carcinoma
+histopathology images, composed of 44 samples digitized with five slide scanners.
+This dataset provides local correspondences between images and thereby isolates
+the scanner-induced domain shift from other inherent, e.g. morphology-induced
+domain shifts. To highlight scanner differences, we present a detailed evaluation
+of color distributions, sharpness, and contrast of the individual scanner subsets.
+Additionally, to quantify the inherent scanner-induced domain shift, we train a
+tumor segmentation network on each scanner subset and evaluate the performance
+both in- and cross-domain. We achieve a class-averaged in-domain intersection
+over union coefficient of up to 0.86 and observe a cross-domain performance
+decrease of up to 0.38, which confirms the inherent domain shift of the presented
+dataset and its negative impact on the performance of deep neural networks.
+1
+Introduction
+The digitization of histological specimens with dedicated slide scanning systems has
+facilitated machine learning-based image analysis for histopathology. These algorithms
+have since assisted pathologists in a variety of routine tasks, e.g. mitotic figure detection
+[1], for which they even have been able to outperform trained experts in controlled
+settings [1, 2]. Still, their performance is highly dependent on the quality and availabil-
+ity of training data [3] and can deteriorate considerably on a test set where the image
+characteristics differ from the training data [4]. Such differences commonly referred to
+as “domain shift” can originate not only from different staining and tissue preparation
+protocols of different pathology laboratories but also from the digitization of histolog-
+1
+arXiv:2301.04423v1  [eess.IV]  11 Jan 2023
+
+2
+Wilm et al.
+ical specimens with different scanning systems. Especially from a clinical perspective,
+domain-agnostic models are important for generating accurate and reliable predictions.
+Previous work has shown that domain generalization techniques, e.g. domain-
+adversarial training, can help to develop domain-agnostic models [5]. For this, a training
+dataset composed of a wide range of different domains is highly desirable. So far, the
+most extensive publicly available multi-scanner histopathology dataset is the training
+set of the MICCAI MItosis DOmain Generalization (MIDOG) 2021 challenge [2]. The
+dataset consists of 2 mm2-sized cropped regions of 200 breast cancer cases digitized
+with four scanners. However, the cases were divided between the scanners, and perfor-
+mance differences can therefore not solely be attributed to the slide scanner but also
+to the case selection. The Mitos & Atypia dataset [6] is the only public multi-scanner
+histopathology dataset with local image correspondences, i.e. the same case was digi-
+tized with multiple slide scanners, however, with 16 cases and two scanners, its extent
+is limited and it does not leave room for experiments with hold-out test scanners.
+In this work, we present a canine cutaneous histopathology dataset, where each of
+the 44 samples was digitized with five different slide scanning systems. This multi-
+scanner dataset provides local image correspondences, useful for domain generalization
+experiments. Accompanied by an annotation database of 1,243 polygon annotations
+for seven histologic classes (tumor, epidermis, dermis, subcutis, bone, cartilage, and
+a combined class of inflammation and necrosis), this is the first publicly available
+multi-scanner segmentation dataset. For each scanner subset, we provide a detailed
+evaluation of color distributions, sharpness, and contrast. To quantify the extent of
+the scanner-induced domain shift, we performed a technical validation of the dataset
+by training a baseline tumor segmentation algorithm on each single scanner domain
+and then testing the algorithm across all scanners. For some scanners, we observed
+a considerable performance decrease, which highlights the domain shift inherent in
+the dataset. The whole slide images (WSIs) and annotation databases are publicly
+available on Zenodo (https://doi.org/10.5281/zenodo.7418555), and code for
+implementing the baseline architectures can be obtained from our GitHub repository
+(https://github.com/DeepPathology/MultiScanner_SCC).
+2
+Materials and methods
+The dataset presented in this work extends the publicly available CATCH dataset [7], a
+collection of 350 WSIs of seven of the most common canine cutaneous tumor subtypes
+(50 WSIs per subtype). For the CATCH dataset, the specimens were digitized with the
+Aperio ScanScope CS2 (Leica, Germany) at a resolution of 0.25 µm/pixel using a 40 ×
+objective lens. Use of these samples was approved by the local governmental authorities
+(State Office of Health and Social Affairs of Berlin, approval ID: StN 011/20). For the
+multi-scanner dataset, we randomly selected one subtype (squamous cell carcinoma)
+and digitized the samples with four additional slide scanners (see Figure 1):
+• NanoZoomer S210 (Hamamatsu, Japan), 0.22 µm/pixel
+• NanoZoomer 2.0-HT (Hamamatsu, Japan), 0.23 µm/pixel
+• Pannoramic 1000 (3DHISTECH, Hungary), 0.25 µm/pixel
+• Aperio GT 450 (Leica, Germany), 0.26 µm/pixel
+
+Multi-Scanner Histopathology Dataset
+3
+(a) CS2
+(b) NZ210
+(c) NZ2.0
+(d) P1000
+(e) GT450
+Fig. 1. Exemplary region of interest of the multi-scanner dataset.
+Due to severe scanning artifacts in at least one of the scans, six specimens were excluded
+from the dataset, resulting in a total of 220 WSIs (44 samples digitized with five
+scanners each). The CATCH annotation database provides annotations for the individual
+tumor subtypes and six additional skin tissue classes (epidermis, dermis, subcutis,
+bone, cartilage, and a combined class of inflammation and necrosis). We transferred all
+annotations to the other scanners using the WSI registration algorithm by Marzahl et
+al. [8] and visually validated them by overlaying the transformed polygon annotations
+onto the scans. We provide public access to the WSIs on Zenodo (https://doi.
+org/10.5281/zenodo.7418555), licensed under a Creative Commons Attribution
+4.0 International License. However, due to storage restrictions, we have converted them
+to lower-resolution pyramidal TIFFs (4 µm/pixel), which has shown to be adequate for
+training segmentation tasks on the CATCH dataset [7].
+2.1
+Dataset validation
+For each scanner subset, we evaluated the average RGB color distribution, sharpness,
+and contrast. For sharpness estimation, we used the cumulative probability of blur
+detection (CPBD) metric [9], which is a perceptual-based image sharpness metric. It is
+computed via edge detection, followed by a blur estimation at the detected edges. The
+CPBD metric then corresponds to the cumulative probability of blur detection, i.e. the
+percentage of image edges that fall below a threshold of a perceptually noticeable blur.
+For implementation details, we refer to [9]. For the analysis of RGB distributions and
+contrast, we used Otsu’s adaptive thresholding to separate foreground tissue from white
+background. For each image, we calculated the average intensities of the color channels
+𝐼𝑅, 𝐼𝐺, and 𝐼𝐵 in the detected tissue regions. Afterward, we converted the regions to
+grayscale and computed the Michelson contrast [10] 𝐶𝑀 as a measure of global contrast.
+2.2
+Technical validation
+For technical validation of the dataset, we trained a segmentation model on each scan-
+ner domain and tested the algorithm across all scanners. For model development, we
+performed a slide-level split into training (N=30), validation (N=5), and test (N=9)
+cases. We trained a UNet with a ResNet18 encoder pre-trained on ImageNet for the
+segmentation into tumor, non-tumor, and background. For this, we combined all skin
+tissue classes into one non-tumor class and used the automatically detected background
+
+4
+Wilm et al.
+areas to train the background class. We trained the networks on image patches sized
+512 × 512 pixels, extracted at a resolution of 4 µm/pixel. During each epoch, we sampled
+50 patches per WSI within the annotated polygons. Due to a high class imbalance, we
+randomly sampled the polygons with a class-weighting of 10 % background and 45 %
+each of tumor and non-tumor regions. For each scanner, we applied z-score normaliza-
+tion with the training set statistics (mean and standard deviation) and performed data
+augmentation using random flipping, affine transformations, and random lightning and
+contrast change. We used the Adam optimizer and trained the networks with a com-
+bination of cross-entropy and Dice loss. We trained the models with a batch size of
+8 and a cyclic learning rate of 10−4 for 100 epochs, after which we observed model
+convergence. Model selection was guided by the highest intersection over union (mIoU)
+on the validation set.
+3
+Results
+Figure 2 shows the RGB distribution of the non-background areas for the complete
+dataset of 44 WSIs per scanner. The distributions match the exemplary patches in
+Figure 1, where the patches of the Aperio CS2 and the NanoZoomer 210 appear redder,
+which is reflected in a shift of the red pixel distributions to higher values. When looking
+at the distributions of the Aperio GT450, all curves are densely located at the higher color
+component values, which corresponds to the bright appearance of the patch in Figure 1d.
+Table 1 summarizes the channel-wise color averages, sharpness, and contrast of the slide
+scanning systems. These results further underline the visual impression of the patches
+in Figure 1. When calculating the ratio of the red and the blue color channel 𝐼𝑅/𝐼𝐵, the
+NZ210 results in a ratio of 1.12 and the NZ2.0 in a ratio of 1.04, which matches the much
+redder appearance of the NZ210 patch and the bluer appearance of the NZ2.0 patch.
+Overall, the CS2, NZ210, NZ2.0, and P1000 show comparable sharpness and contrast
+values, while the Aperio 450 exhibits a slightly higher sharpness but a considerably lower
+contrast. Figure 3 visualizes the mIoU when training the segmentation network on one
+scanner, and testing it on all scanners. The results show high in-domain performance
+(diagonal) with mIoU values between 0.82 for the P1000 and GT450, and 0.86 for
+the NZ210. The cross-domain performance highlights the scanner-induced domain shift
+inherent in our dataset. While the networks trained on the CS2 and the NZ210 generalize
+considerably well, with performance decreases of up to 0.08 and 0.12 compared to the in-
+domain mIoU, the highest cross-domain performance drop was observed when training
+0
+50
+100
+150
+200
+250
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+Density
+(a) CS2
+0
+50
+100
+150
+200
+250
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+Density
+(b) NZ210
+0
+50
+100
+150
+200
+250
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+Density
+(c) NZ2.0
+0
+50
+100
+150
+200
+250
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+Density
+(d) P1000
+0
+50
+100
+150
+200
+250
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+0.030
+0.035
+0.040
+Density
+(e) GT450
+Fig. 2. Kernel density estimation of RGB values per scanner.
+
+Multi-Scanner Histopathology Dataset
+5
+Tab. 1. Channel-wise color distributions 𝐼𝑅, 𝐼𝐺, and 𝐼𝐵, sharpness 𝑆𝐶𝑃𝐵𝐷 calculated as cumu-
+lative probability of blur detection, and Michelson contrast 𝐶𝑀 of the scanners (𝜇 ± 𝜎).
+.
+𝐼𝑅
+𝐼𝐺
+𝐼𝐵
+𝑆𝐶𝑃𝐵𝐷
+𝐶𝑀
+CS2
+201.84 ± 19.46
+153.18 ± 35.41
+171.54 ± 30.02
+0.80 ± 0.02
+0.74 ± 0.12
+NZ210
+218.88 ± 17.96
+172.64 ± 28.04
+195.26 ± 20.15
+0.82 ± 0.03
+0.81 ± 0.14
+NZ2.0
+192.49 ± 21.63
+153.46 ± 36.72
+184.51 ± 23.90
+0.81 ± 0.02
+0.81 ± 0.13
+P1000
+223.41 ± 18.60
+164.97 ± 41.15
+211.44 ± 21.64
+0.80 ± 0.02
+0.71 ± 0.14
+GT450
+226.59 ± 12.99
+208.18 ± 20.88
+218.80 ± 15.92
+0.84 ± 0.04
+0.53 ± 0.15
+on the P1000, with a decrease of up to 0.38. A visual inspection of segmentation results
+showed that the network trained on the P1000 misclassified many background areas
+of the other scanners. A reason might be the integrated tissue detection of the P1000,
+which sets all pixels outside the tissue bounding box to (255, 255, 255) in order to
+reduce scanning times. This artificially removes common artifacts, e.g. dust particles,
+and the network might only look for high pixel values and not learn the morphological
+characteristics of background areas.
+CS2
+NZ210
+NZ2.0
+P1000
+GT450
+test
+CS2
+NZ210
+NZ2.0
+P1000
+GT450
+train
+0.83
+0.83
+0.82
+0.75
+0.8
+0.79
+0.86
+0.84
+0.74
+0.81
+0.71
+0.7
+0.84
+0.82
+0.79
+0.6
+0.44
+0.65
+0.82
+0.55
+0.81
+0.71
+0.81
+0.7
+0.82
+0.45
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+0.85
+Fig. 3. Scanner-wise performance of segmentation net-
+works. Matrix entry 𝑚𝑖, 𝑗 is the mean intersection over
+union (mIoU) when training on the scanning system
+in row 𝑖 and testing on the scanning system in column
+𝑗. Diagonal elements indicate in-domain performance,
+whereas off-diagonal elements represent cross-domain
+performance.
+4
+Discussion
+Our experiments have demonstrated the negative impact of scanner-induced domain
+shifts on the performance of deep neural networks, indicated by a considerable de-
+crease in mIoU on unseen scanners. This confirms the observations of previous works
+and supports the need for methods that can tackle this domain shift and adequate
+datasets to evaluate their generalization capability. The presented dataset exceeds exist-
+ing multi-scanner datasets in terms of sample size and scanning systems. Furthermore,
+it provides local image correspondences, which isolate the scanner-induced from the
+morphology-induced domain shift and allow the development of algorithms dependent
+on these correspondences, e.g. WSI registration algorithms. We have implicitly shown
+the eligibility of our dataset for this application by successfully transferring our anno-
+tation database from the CS2 scanner to the remaining scanner using WSI registration.
+The detailed evaluation of our scanner subsets has highlighted considerable differences
+
+6
+Wilm et al.
+regarding color distributions and contrasts present in clinically used scanners. Surpris-
+ingly, even though our evaluations resulted in the lowest contrast value for the Aperio
+GT450, this did not impede segmentation performance, shown by an in-domain mIoU
+of 0.82, which is comparable to the in-domain mIoUs of the remaining scanners. In our
+technical validation, we observed a large cross-domain performance decrease, especially
+when training on the P1000 scanner. We assume that this can mainly be attributed to the
+unique pre-processing steps of the scanner vendor, as the P1000 showed similar image
+statistics to the CS2 but their average cross-domain performance differed considerably.
+However, we also observed a cross-domain performance decrease for the remaining
+scanners, which indicates that some of the learned feature representations did not gen-
+eralize well across scanners. Future work could focus on a closer evaluation of which
+scanner characteristics hinder the extraction of domain-agnostic features and should
+therefore be disregarded, e.g. by using specific filters for data pre-processing or using
+adversarial training to punish the extraction of these features.
+Acknowledgement. F.W. gratefully acknowledges the financial support received by
+Merck Healthcare KGaA and the technical support received by the Clinical Assay
+Strategy 1 group at Merck Healthcare KGaA during sample digitization. K.B. gratefully
+acknowledges support by d.hip campus - Bavarian aim in form of a faculty endowment.
+References
+1. Aubreville M, Bertram CA, Marzahl C, Gurtner C, Dettwiler M, Schmidt A et al. Deep
+learning algorithms out-perform veterinary pathologists in detecting the mitotically most
+active tumor region. Sci Rep 10:16447 (2020), pp. 1–11.
+2. Aubreville M, Stathonikos N, Bertram CA, Klopleisch R, Hoeve N ter, Ciompi F et al. Mitosis
+domain generalization in histopathology images–The MIDOG challenge. Med Image Anal
+84:102699 (2023).
+3. Deng S, Zhang X, Yan W, Chang EI, Fan Y, Lai M et al. Deep learning in digital pathology
+image analysis: a survey. Front Med 14.4 (2020), pp. 470–487.
+4. Stacke K, Eilertsen G, Unger J, Lundström C. Measuring domain shift for deep learning in
+histopathology. IEEE J Biomed Health Inform 25.2 (2020), pp. 325–336.
+5. Wilm F, Marzahl C, Breininger K, Aubreville M. Domain adversarial RetinaNet as a refer-
+ence algorithm for the MItosis DOmain Generalization challenge. Biomedical Image Regis-
+tration, Domain Generalisation and Out-of-Distribution Analysis: MICCAI 2021 Challenges.
+Springer. 2022, pp. 5–13.
+6. Roux L, Racoceanu D, Capron F, Calvo J, Attieh E, Le Naour G et al. Mitos & Atypia. Image
+Pervasive Access Lab (IPAL), Agency Sci., Technol. & Res. Inst. Infocom Res., Singapore,
+Tech. Rep 1 (2014), pp. 1–8.
+7. Wilm F et al. CAnine CuTaneous Cancer Histology dataset (version 1). The Cancer Imaging
+Archive (2022). https://doi.org/10.7937/TCIA.2M93-FX66.
+8. Marzahl C, Wilm F, F. DF, Tharun L, Perner S, Bertram CA et al. Robust quad-tree based
+registration on whole slide images. Comput Pathol (2021). PMLR, 2021, pp. 181–190.
+9. Narvekar ND, Karam LJ. A no-reference image blur metric based on the cumulative proba-
+bility of blur detection (CPBD). IEEE Trans Image Process 20.9 (2011), pp. 2678–2683.
+10. Michelson AA. Studies in optics. Courier Corporation, 1995.
+

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+page_content=' Germany  †shared senior authors  frauke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
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+page_content='de  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' In histopathology, scanner-induced domain shifts are known to impede  the performance of trained neural networks when tested on unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Multi-  domain pre-training or dedicated domain-generalization techniques can help to  develop domain-agnostic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For this, multi-scanner datasets with a high  variety of slide scanning systems are highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We present a publicly  available multi-scanner dataset of canine cutaneous squamous cell carcinoma  histopathology images, composed of 44 samples digitized with five slide scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
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+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
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+page_content=' To highlight scanner differences, we present a detailed evaluation  of color distributions, sharpness, and contrast of the individual scanner subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  Additionally, to quantify the inherent scanner-induced domain shift, we train a  tumor segmentation network on each scanner subset and evaluate the performance  both in- and cross-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We achieve a class-averaged in-domain intersection  over union coefficient of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='86 and observe a cross-domain performance  decrease of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='38, which confirms the inherent domain shift of the presented  dataset and its negative impact on the performance of deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  1  Introduction  The digitization of histological specimens with dedicated slide scanning systems has  facilitated machine learning-based image analysis for histopathology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' These algorithms  have since assisted pathologists in a variety of routine tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' mitotic figure detection  [1], for which they even have been able to outperform trained experts in controlled  settings [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Still, their performance is highly dependent on the quality and availabil-  ity of training data [3] and can deteriorate considerably on a test set where the image  characteristics differ from the training data [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Such differences commonly referred to  as “domain shift” can originate not only from different staining and tissue preparation  protocols of different pathology laboratories but also from the digitization of histolog-  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='04423v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='IV]  11 Jan 2023  2  Wilm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  ical specimens with different scanning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Especially from a clinical perspective,  domain-agnostic models are important for generating accurate and reliable predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  Previous work has shown that domain generalization techniques, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' domain-  adversarial training, can help to develop domain-agnostic models [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For this, a training  dataset composed of a wide range of different domains is highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' So far, the  most extensive publicly available multi-scanner histopathology dataset is the training  set of the MICCAI MItosis DOmain Generalization (MIDOG) 2021 challenge [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The  dataset consists of 2 mm2-sized cropped regions of 200 breast cancer cases digitized  with four scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' However, the cases were divided between the scanners, and perfor-  mance differences can therefore not solely be attributed to the slide scanner but also  to the case selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The Mitos & Atypia dataset [6] is the only public multi-scanner  histopathology dataset with local image correspondences, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' the same case was digi-  tized with multiple slide scanners, however, with 16 cases and two scanners, its extent  is limited and it does not leave room for experiments with hold-out test scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  In this work, we present a canine cutaneous histopathology dataset, where each of  the 44 samples was digitized with five different slide scanning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' This multi-  scanner dataset provides local image correspondences, useful for domain generalization  experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Accompanied by an annotation database of 1,243 polygon annotations  for seven histologic classes (tumor, epidermis, dermis, subcutis, bone, cartilage, and  a combined class of inflammation and necrosis), this is the first publicly available  multi-scanner segmentation dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For each scanner subset, we provide a detailed  evaluation of color distributions, sharpness, and contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' To quantify the extent of  the scanner-induced domain shift, we performed a technical validation of the dataset  by training a baseline tumor segmentation algorithm on each single scanner domain  and then testing the algorithm across all scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For some scanners, we observed  a considerable performance decrease, which highlights the domain shift inherent in  the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The whole slide images (WSIs) and annotation databases are publicly  available on Zenodo (https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='7418555), and code for  implementing the baseline architectures can be obtained from our GitHub repository  (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='com/DeepPathology/MultiScanner_SCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  2  Materials and methods  The dataset presented in this work extends the publicly available CATCH dataset [7], a  collection of 350 WSIs of seven of the most common canine cutaneous tumor subtypes  (50 WSIs per subtype).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For the CATCH dataset, the specimens were digitized with the  Aperio ScanScope CS2 (Leica, Germany) at a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='25 µm/pixel using a 40 ×  objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Use of these samples was approved by the local governmental authorities  (State Office of Health and Social Affairs of Berlin, approval ID: StN 011/20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For the  multi-scanner dataset, we randomly selected one subtype (squamous cell carcinoma)  and digitized the samples with four additional slide scanners (see Figure 1):  NanoZoomer S210 (Hamamatsu, Japan), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='22 µm/pixel  NanoZoomer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0-HT (Hamamatsu, Japan), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='23 µm/pixel  Pannoramic 1000 (3DHISTECH, Hungary), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='25 µm/pixel  Aperio GT 450 (Leica, Germany), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='26 µm/pixel  Multi-Scanner Histopathology Dataset  3  (a) CS2  (b) NZ210  (c) NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0  (d) P1000  (e) GT450  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Exemplary region of interest of the multi-scanner dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  Due to severe scanning artifacts in at least one of the scans, six specimens were excluded  from the dataset, resulting in a total of 220 WSIs (44 samples digitized with five  scanners each).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The CATCH annotation database provides annotations for the individual  tumor subtypes and six additional skin tissue classes (epidermis, dermis, subcutis,  bone, cartilage, and a combined class of inflammation and necrosis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We transferred all  annotations to the other scanners using the WSI registration algorithm by Marzahl et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' [8] and visually validated them by overlaying the transformed polygon annotations  onto the scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We provide public access to the WSIs on Zenodo (https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='7418555), licensed under a Creative Commons Attribution  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' However, due to storage restrictions, we have converted them  to lower-resolution pyramidal TIFFs (4 µm/pixel), which has shown to be adequate for  training segmentation tasks on the CATCH dataset [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='1  Dataset validation  For each scanner subset, we evaluated the average RGB color distribution, sharpness,  and contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For sharpness estimation, we used the cumulative probability of blur  detection (CPBD) metric [9], which is a perceptual-based image sharpness metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' It is  computed via edge detection, followed by a blur estimation at the detected edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The  CPBD metric then corresponds to the cumulative probability of blur detection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' the  percentage of image edges that fall below a threshold of a perceptually noticeable blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  For implementation details, we refer to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For the analysis of RGB distributions and  contrast, we used Otsu’s adaptive thresholding to separate foreground tissue from white  background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For each image, we calculated the average intensities of the color channels  𝐼𝑅, 𝐼𝐺, and 𝐼𝐵 in the detected tissue regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Afterward, we converted the regions to  grayscale and computed the Michelson contrast [10] 𝐶𝑀 as a measure of global contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='2  Technical validation  For technical validation of the dataset, we trained a segmentation model on each scan-  ner domain and tested the algorithm across all scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For model development, we  performed a slide-level split into training (N=30), validation (N=5), and test (N=9)  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We trained a UNet with a ResNet18 encoder pre-trained on ImageNet for the  segmentation into tumor, non-tumor, and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For this, we combined all skin  tissue classes into one non-tumor class and used the automatically detected background  4  Wilm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  areas to train the background class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We trained the networks on image patches sized  512 × 512 pixels, extracted at a resolution of 4 µm/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' During each epoch, we sampled  50 patches per WSI within the annotated polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Due to a high class imbalance, we  randomly sampled the polygons with a class-weighting of 10 % background and 45 %  each of tumor and non-tumor regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' For each scanner, we applied z-score normaliza-  tion with the training set statistics (mean and standard deviation) and performed data  augmentation using random flipping, affine transformations, and random lightning and  contrast change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We used the Adam optimizer and trained the networks with a com-  bination of cross-entropy and Dice loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We trained the models with a batch size of  8 and a cyclic learning rate of 10−4 for 100 epochs, after which we observed model  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Model selection was guided by the highest intersection over union (mIoU)  on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  3  Results  Figure 2 shows the RGB distribution of the non-background areas for the complete  dataset of 44 WSIs per scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The distributions match the exemplary patches in  Figure 1, where the patches of the Aperio CS2 and the NanoZoomer 210 appear redder,  which is reflected in a shift of the red pixel distributions to higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' When looking  at the distributions of the Aperio GT450, all curves are densely located at the higher color  component values, which corresponds to the bright appearance of the patch in Figure 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  Table 1 summarizes the channel-wise color averages, sharpness, and contrast of the slide  scanning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' These results further underline the visual impression of the patches  in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' When calculating the ratio of the red and the blue color channel 𝐼𝑅/𝐼𝐵, the  NZ210 results in a ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='12 and the NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0 in a ratio of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='04, which matches the much  redder appearance of the NZ210 patch and the bluer appearance of the NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0 patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  Overall, the CS2, NZ210, NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0, and P1000 show comparable sharpness and contrast  values, while the Aperio 450 exhibits a slightly higher sharpness but a considerably lower  contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Figure 3 visualizes the mIoU when training the segmentation network on one  scanner, and testing it on all scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The results show high in-domain performance  (diagonal) with mIoU values between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='82 for the P1000 and GT450, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='86 for  the NZ210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The cross-domain performance highlights the scanner-induced domain shift  inherent in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' While the networks trained on the CS2 and the NZ210 generalize  considerably well, with performance decreases of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='08 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='12 compared to the in-  domain mIoU, the highest cross-domain performance drop was observed when training  0  50  100  150  200  250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='040  Density  (a) CS2  0  50  100  150  200  250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='040  Density  (b) NZ210  0  50  100  150  200  250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='040  Density  (c) NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0  0  50  100  150  200  250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='040  Density  (d) P1000  0  50  100  150  200  250  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='030  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='040  Density  (e) GT450  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Kernel density estimation of RGB values per scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  Multi-Scanner Histopathology Dataset  5  Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Channel-wise color distributions 𝐼𝑅, 𝐼𝐺, and 𝐼𝐵, sharpness 𝑆𝐶𝑃𝐵𝐷 calculated as cumu-  lative probability of blur detection, and Michelson contrast 𝐶𝑀 of the scanners (𝜇 ± 𝜎).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  𝐼𝑅  𝐼𝐺  𝐼𝐵  𝑆𝐶𝑃𝐵𝐷  𝐶𝑀  CS2  201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='84 ± 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='46  153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='18 ± 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='41  171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='54 ± 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='12  NZ210  218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='88 ± 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='96  172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='64 ± 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='04  195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='26 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='14  NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0  192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='49 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='63  153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='46 ± 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='72  184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='51 ± 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='81 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='13  P1000  223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='41 ± 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='60  164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='97 ± 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='15  211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='44 ± 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='14  GT450  226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='59 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='99  208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='18 ± 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='88  218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='80 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='15  on the P1000, with a decrease of up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' A visual inspection of segmentation results  showed that the network trained on the P1000 misclassified many background areas  of the other scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' A reason might be the integrated tissue detection of the P1000,  which sets all pixels outside the tissue bounding box to (255, 255, 255) in order to  reduce scanning times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' This artificially removes common artifacts, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' dust particles,  and the network might only look for high pixel values and not learn the morphological  characteristics of background areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  CS2  NZ210  NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0  P1000  GT450  test  CS2  NZ210  NZ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='0  P1000  GT450  train  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
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+page_content='85  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Scanner-wise performance of segmentation net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Matrix entry 𝑚𝑖, 𝑗 is the mean intersection over  union (mIoU) when training on the scanning system  in row 𝑖 and testing on the scanning system in column  𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Diagonal elements indicate in-domain performance,  whereas off-diagonal elements represent cross-domain  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  4  Discussion  Our experiments have demonstrated the negative impact of scanner-induced domain  shifts on the performance of deep neural networks, indicated by a considerable de-  crease in mIoU on unseen scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' This confirms the observations of previous works  and supports the need for methods that can tackle this domain shift and adequate  datasets to evaluate their generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The presented dataset exceeds exist-  ing multi-scanner datasets in terms of sample size and scanning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Furthermore,  it provides local image correspondences, which isolate the scanner-induced from the  morphology-induced domain shift and allow the development of algorithms dependent  on these correspondences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' WSI registration algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We have implicitly shown  the eligibility of our dataset for this application by successfully transferring our anno-  tation database from the CS2 scanner to the remaining scanner using WSI registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  The detailed evaluation of our scanner subsets has highlighted considerable differences  6  Wilm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  regarding color distributions and contrasts present in clinically used scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Surpris-  ingly, even though our evaluations resulted in the lowest contrast value for the Aperio  GT450, this did not impede segmentation performance, shown by an in-domain mIoU  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='82, which is comparable to the in-domain mIoUs of the remaining scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' In our  technical validation, we observed a large cross-domain performance decrease, especially  when training on the P1000 scanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' We assume that this can mainly be attributed to the  unique pre-processing steps of the scanner vendor, as the P1000 showed similar image  statistics to the CS2 but their average cross-domain performance differed considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  However, we also observed a cross-domain performance decrease for the remaining  scanners, which indicates that some of the learned feature representations did not gen-  eralize well across scanners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Future work could focus on a closer evaluation of which  scanner characteristics hinder the extraction of domain-agnostic features and should  therefore be disregarded, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' by using specific filters for data pre-processing or using  adversarial training to punish the extraction of these features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' gratefully acknowledges the financial support received by  Merck Healthcare KGaA and the technical support received by the Clinical Assay  Strategy 1 group at Merck Healthcare KGaA during sample digitization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' gratefully  acknowledges support by d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='hip campus - Bavarian aim in form of a faculty endowment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
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+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
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+page_content=' Domain adversarial RetinaNet as a refer-  ence algorithm for the MItosis DOmain Generalization challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Biomedical Image Regis-  tration, Domain Generalisation and Out-of-Distribution Analysis: MICCAI 2021 Challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
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+page_content=' 5–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Roux L, Racoceanu D, Capron F, Calvo J, Attieh E, Le Naour G et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Mitos & Atypia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Image  Pervasive Access Lab (IPAL), Agency Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=', Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' & Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Infocom Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=', Singapore,  Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Rep 1 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
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+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Wilm F et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' CAnine CuTaneous Cancer Histology dataset (version 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' The Cancer Imaging  Archive (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='7937/TCIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='2M93-FX66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Marzahl C, Wilm F, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' DF, Tharun L, Perner S, Bertram CA et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Robust quad-tree based  registration on whole slide images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Comput Pathol (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' PMLR, 2021, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' 181–190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Narvekar ND, Karam LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' A no-reference image blur metric based on the cumulative proba-  bility of blur detection (CPBD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' IEEE Trans Image Process 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='9 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' 2678–2683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Michelson AA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Studies in optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}
+page_content=' Courier Corporation, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE3T4oBgHgl3EQfRgnj/content/2301.04423v1.pdf'}

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+CageCoach: Sharing-Oriented Redaction-Capable
+Distributed Cryptographic File System
+Jason Carpenter
+CARPE415@umn.edu
+University of Minnesota
+Minneapolis, MN
+Zhi-Li Zhang
+zhzhang@cs.umn.edu
+University of Minnesota
+Minneapolis, MN
+ABSTRACT
+The modern data economy is built on sharing data. However,
+sharing data can be an expensive and risky endeavour. Exist-
+ing sharing systems like Distributed File Systems provide full
+read, write, and execute Role-based Access Control (RBAC)
+for sharing data, but can be expensive and difficult to scale.
+Likewise such systems operate on a binary access model for
+their data, either a user can read all the data or read none of
+the data. This approach is not necessary for a more read-only
+oriented data landscape, and one where data contains many
+dimensions that represent a risk if overshared. In order to
+encourage users to share data and smooth out the process
+of accessing such data a new approach is needed. This new
+approach must simplify the RBAC of older DFS approaches
+to something more read-only and something that integrates
+redaction for user protections.
+To accomplish this we present CageCoach, a simple sharing-
+oriented Distributed Cryptographic File System (DCFS). Cage-
+Coach leverages the simplicity and speed of basic HTTP,
+linked data concepts, and automatic redaction systems to
+facilitate safe and easy sharing of user data. The implemen-
+tation of CageCoach is available at https://github.umn.edu/
+CARPE415/CageCoach.
+1
+INTRODUCTION
+User-generated data drives the modern world. Everything
+from Uber driver rides and Google search queries to video
+game experiences and Amazon purchase patterns feed user
+data back into these systems to provide insights for improve-
+ment. Additionally, users sharing their data as part of crowd
+sourcing solutions has proven key to reverse engineering gig
+working applications such as Uber[5, 16, 18], Lyft[16, 18],
+and Shipt[3, 18]. Further these efforts help solve civic and
+national needs such as with Atlanta’s Data Dashboard[13],
+Minneapolis’s Opendata program[7], or the United State’s
+Citizen Science initiative[8].
+However, users providing their data to these initiatives
+often comes with a level of risk and a loss of control over the
+data they provide. Once a user has handed over information
+the safety considerations, redaction approaches, and man-
+agement decisions are out of their control. Further, should
+any shared user data become dangerous to a user, the user
+has no more sway to alleviate this risk other than ask the
+current data holder to act, a practice often fruitless.
+In order to further encourage users to share their data, a
+new sharing oriented data hosting system is required. Such a
+platform must be simple to implement, easy to request data
+from, but still provide some assurances of privacy and safety
+for users involved. Crucially it should remain in the user’s
+control, and not be subject to control by others even those
+hosting data such as on public hosting systems. The privacy
+capability must be granular not just in who can access data
+but what specific data is accessible. For example, for some
+users, sharing their full name to everyone who asks is un-
+reasonable. Thus they should be able to share with some a
+partial redaction of their name. Existing works such as Dis-
+tributed File Systems (DFS) are promising, but require exten-
+sive implementation, Role-based Access Control (RBAC) en-
+forcement, and do not implement granular redaction. Other
+platforms like Google Drive, Dropbox, and Kaggle are great
+for sharing bulk data but also do not provide granular redac-
+tion and require trusting of the platform holders to not share
+otherwise redacted user data.
+In this work, we introduce CageCoach a sharing oriented
+distributed cryptographic file system. CageCoach’s notable
+features are:
+• Simple Trustless DCFS built over HTTP GET/POST
+• Customizable RBAC and Datatype Granular Redac-
+tion Pipeline
+• Easier sharing with Decentralized data access and
+centralized user control
+CageCoach streamlines the older RBAC based models of
+DFSs and decentralizes the data hosting approaches of plat-
+forms making for an overall simpler means of sharing data
+with others while retaining granular privacy control for users.
+This system is leverages simple HTTP GET/POST operations
+to interact with symmetrically encrypted files hosted on any
+HTTP platform to achieve decentralized hosting. These files
+point back to their owners, represented by a controlling
+server, that can facilitate redacted data access for a data re-
+quester providing user control of data access. Finally, the
+1
+arXiv:2301.04214v1  [cs.CR]  10 Jan 2023
+
+,
+user’s controlling server applies user defined redaction oper-
+ations from a suite of modules CageCoach provides to reduce
+sensitive data leakage.
+CageCoach’s code can be found at https://github.umn.edu/
+CARPE415/CageCoach.
+2
+RELATED WORK
+Distributed File Systems (DFS) and cryptographic file sys-
+tems (DCFS) have been around for a long time with some
+works as early as 1993[1] and as recent as 2020[2]. These are
+mature fields with well-developed and commercial products
+we see every day, such as Dropbox, GoogleDrive, Hadoop,
+Ceph, and others[22]. Despite this, the changing data land-
+scape and changing usage behaviors with data invite re-
+examinations of existing systems to better fit them for a new
+era. The work must relevant in the current data landscape,
+data redaction, is an old field but with a renewed interest in
+the face of big data breaches, data privacy concerns, and ma-
+chine learning for data protection. In this section we outline
+these two related areas and contrast them with our proposed
+system.
+2.1
+Distributed & Cryptographic File
+Systems
+Distributed File Systems (DFS) are systems for maintaining
+coherent file management across desperate hosting devices.
+Examples include standard file hosting such as Google Drive,
+Dropbox, and InRupt’s Solid[19]. Such systems have a long
+history and continued relevance in the modern era. DFS also
+manifest as cloud storage systems, albeit with looser file sys-
+tem format adherence to mesh with the more diverse Internet
+access environment. Extending DFSs into privacy and secu-
+rity oriented spaces yields the Distributed Cryptographic
+File System (DCFS) domain. Works such as UPSS[2] focus
+on creating a sharing-oriented and protective DFS with full
+RBAC and mutable verifiable histories of each file involved as
+a check against malicious behavior. Further other works such
+as [10, 12] aim to utilize the blockchain to achieve the same
+RBAC with a more decentralized approach. Finally, other
+approaches aim to refine key management in encryption for
+DFS[14].These systems while powerful, rely on relatively
+expensive RBAC and infrastructure or require significant
+trust for the platform holders. In the former case, simplify-
+ing the RBAC with the mostly read-only reality of user data
+can lower RBAC complexity significantly. In the latter case,
+hosting infrastructure is still necessary, but one must create
+a trustless environment in order to retain control of one’s
+data even on such hosting platforms.
+Our work focuses on streamlining data sharing by creat-
+ing a middlepoint between strong, rigid, and RBAC focused
+approaches such as DCFSs and trust-oriented data platforms
+and services like Uber, Kaggle, and Gridwise.
+2.2
+Data Redaction
+Data redaction is not a new field, but has gained vigor in the
+last decade or so as the data economy has shaped. Redac-
+tion provides the means for which sensitive data can be
+made less sensitive and thus less dangerous in the event of
+leaks, breaches, or theft. Likewise, redaction has its place in
+academic publications when such publications may contain
+in themselves dangerous or sensitive information[4]. Many
+existing tools provide a user the quick means of redacting
+a document such as [6] and [20]. A handful of commercial
+products, such as [21], [15], and [17], apply machine learning
+to identify and remove automatically sensitive data. Finally,
+other work such as [11] highlight an interesting scenario
+where redaction itself must be transparent enough such that
+the redaction doesn’t mislead the information. These systems
+as implemented are not part of a sharing pipeline and are
+applied ad-hoc to data. A system such as the one outlined by
+UPSS[2], envisions such technologies are part of a pipeline
+of data requests but did not implement or specify beyond
+such designs.
+Our work applies the concepts behind these redaction
+systems, but crucially, as part of a standard granular access
+pipeline and not as a one-off and static redaction. This in
+effect realizes some aspects of the UPSS[2] pipeline, but with-
+out the more complex full RBAC suite.
+3
+PROBLEM AND DESIGN GOALS
+In order to build a system that encourages users to share their
+data two primary problems and design considerations must
+be achieved: Simplification of access control for accessing
+and requesting data and automatic policy informed data
+redaction. With these two aspects a sharing-oriented DFS
+will lower the cost of sharing and accessing data and provide
+a wide net of protections for users who choose to share.
+3.1
+Simplify Access Control For Data
+Existing DFS systems utilize a full suite of RBAC function-
+ality to provide read, write, and execute functionality for
+shared files. These provisions while useful, require signif-
+icant infrastructure such as certificates and user profiles
+registered within the computational structure of the data
+host. This full suite of RBAC is necessary if the group of
+users intended to read, write, and/or execute the shared data,
+but costly if sharing (read only) is the intention. By removing
+the write and execute assumptions of RBAC we can in turn
+simplify the operating infrastructure required for accessing
+data and making sharing a lower cost effort. This lower cost
+is necessary for encouraging users to share their data, as it
+2
+
+CageCoach: Sharing-Oriented Redaction-Capable Distributed Cryptographic File System
+,
+will be easier to host for consumption, and for consumers of
+data as it will be easier to access.
+3.2
+Provide Integrated Automatic User
+Data Redaction
+Regardless of ease of access, users must be given some as-
+surances of safety, privacy, and proper use for their data.
+Traditional RBAC focuses on binary access models for data,
+either a user can read all the data or none of the data in a typi-
+cally hosted file. This approach is not adequate for data items
+that contain core sensitive fields. For example, a typical sales
+receipt is useful for inventory systems and market trending
+services, as they provide insights into purchases and sales
+trends, however, these same receipts may contain the pur-
+chaser’s name, credit card information, and/or address and
+location. Such fields are not important for the overall trend,
+but present a security risk for the user. In a binary RBAC
+model, such fields would available if the receipt is available.
+A more granular approach to access is needed. Such an ap-
+proach is outlined but not realized or specified by UPSS[2].
+Such an approach would require that when a user’s data is re-
+quested by another, a trusted middle system acquires the raw
+full set of data, and then redacts and removes information
+that is included in the data but not allowed for that partic-
+ular user. For example, removing the name, address, and
+credit fields from the sales receipt scenario. This approach
+is required to provide granular and safer exposure of user’s
+data for general consumption. Further, this process can be
+handled by user-defined policy thus providing guidelines
+for any user data added in the future thus lowering sharing
+costs further.
+4
+CAGECOACH SYSTEM
+We realize the goals of a sharing-oriented DFS with our
+system CageCoach. CageCoach simplifies the RBAC and
+infrastructure of existing DFSs and integrates redaction tech-
+nologies into a data request pipeline. All of this together
+creates a simple and easy means for users to safely and eas-
+ily share their data. CageCoach is organized around several
+concepts and a flow, outlined in fig. 1. Requesters, who re-
+quest user data. Data hosts, which host encrypted data files
+and some attached meta data files. Finally, a Data Control
+Server (DCS) which manages the owner’s data, processes
+requests made by requesters, and redacts outgoing sensitive
+data. CageCoach’s operational use-case is:
+(1) A owner uploads some data (video, text, audio, etc) to
+a hosting system after encrypting and creating a meta
+file for the data.
+(2) A requester sees this data and examines the meta file
+(using GET for example) for information as to where
+the owner’s DCS operates.
+Figure 1: CageCoach System, providing a streamlined
+means for requestors to ask for data and receive useful
+but protected data.
+(3) The requester sends a POST request to the owner’s
+DCS server, asking to view the original data item.
+(4) The DCS receives this request, verifies the requester’s
+identity through asymmetric key phrase decryption,
+and then uses GET to retrieve the remotely hosted
+encrypted data file.
+(5) The DCS decrypts the file with its own internal sym-
+metric key and then applies a series of redaction oper-
+ations on the data.
+(6) The DCS forwards the remaining unredacted data to
+the requester, completing the request and preventing
+unnecessary or forbidden data from leaving encrypt-
+ed/controlled space.
+The details for how the RBAC is simplified and how the
+redaction is integrated is detailed in the following sections.
+4.1
+Simplifying RBAC Using HTTP And
+Read-Only Assumptions
+CageCoach simplifies the primary RBAC and infrastructure
+of other DFSs by assuming that user data need only be read,
+not written too or executed collaboratively. Additionally,
+unlike UPSS[2], since there is no write permissions data
+versions are no longer necessary thus can relax the assump-
+tion UPSS makes for needing a transparent modifications
+tree. With this simplification in mind, CageCoach utilizes
+the most common means of read-only operation on the Inter-
+net: HTTP GET. This means that user data can be hosted on
+any system that facilitates HTTP GET, such as open source
+systems like Apache2. The data that gets hosted is the user’s
+encrypted file and a plain text meta data file. Using some con-
+cepts of linked data, the meta data file points to the owner’s
+DCS to actually facilitate the request for data among other
+fields. The total definition for this meta data file is:
+3
+
+(2)Directrequestertodataowner
+B
+(1)Requestaccesstodata
+000
+(3) Downloads encrypted file
+Data Host
+(Dropbox,GDrive,Apache)
+000
+000
+HTTP Data Control
+Requester
+Server (Dcs)
+AccessControl(ACL)And
+DataCensoringRules(DCR)
+(5)Alloweddatais returned
+(4)Decryptsandprocessesfile,
+• owner-url: URL indicating where the owner’s DCS is. The
+place where any request will be processed.
+• meta-data: User filled info tags about the data, such as
+what format it is, overall context. All of this information is
+optional.
+• description: A more textual description of the data, op-
+tional if an owner wishes to provide more than just tags of
+information.
+• data-url: The URL indicating where the data this meta file
+belongs to is. This is important for providing some backup if
+the meta file is moved elsewhere or if it must live elsewhere
+in hosting.
+• data-hash-sha1: A sha1 of the encrypted file to provide a
+minimal check for any requester that wishes to double check
+the file they are asking about.
+Despite our overall read-only approach, some computa-
+tional efforts are still required. Namely the decryption of
+the requested file and the granular redaction of information
+within this file. The purpose of redirecting the requester
+from the data host is to provide a centralized response by
+the owner and the computational space for redaction poli-
+cies. The requester will send an HTTP POST request to the
+DCS indicated by the owner-url and receive a decrypted
+and redacted data file. The DCS’s process is implemented as
+a basic python HTTP server. The process involves several
+steps: 1) Receive a POST request with the URL of the data
+being requested and optionally an ID and asymmetrically
+encrypted phrase to verify the requester’s identity. Cage-
+Coach implements this with RSA public/private key pairs.
+2) Locate the data profile for the requested data on the DCS
+server, itself a simple text file containing pointers to decrypt
+and identify the requested data. Additionally, if the user is
+registered with the DCS (registry comprised of a private key
+for decrypting phrases, the plain text passphrase, and a id
+name) it will load their profile. We implement this as simply
+a separate json file containing each requester’s information.
+Our approach assumes this registry happens outside of the
+CageCoach architecture but can utilize it. 3) The DCS will
+download the encrypted file from its host using HTTP GET.
+After reception, the DCS will decrypt the data file and load
+the redaction policies that match the specific data item (by
+its name), the data type (json, mp3, etc), and finally the poli-
+cies for the requester (if provided). CageCoach implements
+this encryption with symmetric keys using pythons Fernet
+library. 4) The DCS will apply these redaction operations,
+gradually chipping away data until left with whatever is al-
+lowed to pass. 5) The remaining data is sent to the requester
+in the POST response. The specifics of how the redaction is
+applied is outlined in the next section.
+Figure 2: CageCoach Redaction Pipeline, providing a
+generalized measure of privacy assurance.
+4.2
+Access Control and Redaction
+Pipelines
+CageCoach’s read-only assumption for user data is not a
+binary, like older models of RBAC based system, but granu-
+lar. By using a series of redaction operations over requested
+data, CageCoach can allow partial access to data. These op-
+erations, dividable by datatype as outlined in Fig. 2, provide
+for blurring faces in images, redacting text in jsons and csvs,
+and muting specific words or background noises recognized
+in audio. In the overall data request pipeline after a user
+has requested data and the DCS has downloaded the target
+data, it will apply these redaction operations according to
+the specific user, datatype, and data item. This provides three
+levels of granularity for controlling data flow outwards to re-
+questers: by datatype (all jsons, csvs, mp3s, etc), by data item
+(ex: specific files like example-1.json hosted on Google Drive
+or example-2.json hosted on dropbox), and by requester id
+(ex: John Doe can access the user’s name, but Jane Doe can
+only see the user’s first name). However, such operations
+that would be specific to an owner, such as blurring only
+the owner’s face, require the owner provide their own data
+to the redacting DCS. Our implementation we provide does
+general redaction such as blurring all faces and removing a
+handful of well known text fields such as social security and
+street addresses. We do not implement an audio redaction
+approach as there isn’t a general python capable pre-built
+audio redaction library nor a common set of what "words"
+should be auto removed, unlike faces in images. CageCoach
+does support extensions to these operations to tailor to spe-
+cific users. Our implementation uses the Haar cascade and
+OpenCV2 [9] python libraries for blurring faces (illustrated
+with the blurring of photo of American Union Army General
+Benjamin Butler fig. 3), and python Pandas to redact textual
+data (example of such in fig. 4).
+4
+
+010110
+RedactionProcessesByDataTypes
+010001
+111011
+Blob
+Age:43
+Name:"JohnDoe"
+SSN:"999-99-9999"
+File
+CensorFields
+Encrypted
+8871
+File
+Image
+CensorFaces/Persons
+4)
+Audio
+CensorAudio SegmentsCageCoach: Sharing-Oriented Redaction-Capable Distributed Cryptographic File System
+,
+Figure 3: CageCoach Redaction Pipeline example blur-
+ring a specific image.
+Figure 4: CageCoach Redaction Pipeline example
+redacting specific text and fields.
+5
+CONCLUSION
+In this work, we introduced a new sharing oriented imple-
+mentation of DCFS: CageCoach. CageCoach streamlines the
+older RBAC heavy and trust-necessary hosting models of
+DFS, while using the simpler HTTP GET/POST ecosystem to
+facilitate easier data sharing. All of this is possible while still
+respecting the privacy of users through granular customize-
+able redaction pipelines that handle removal of sensitive user
+information.
+6
+LIMITATIONS AND FUTURE WORK
+CageCoach has a set of drawbacks and limitations. Cage-
+Coach is implemented as a demonstration of a new inter-
+pretation of sharing-oriented DCFS and not intended for
+industrial or commercial use. Future implementations would
+need to provide better integration with hosting services like
+Google and Dropbox, and provide tougher and more robust
+security checks and infrastructure. Likewise future work
+improvements would be needed to make the redaction oper-
+ations more capable and workable on a wider set of diverse
+data. Notably there are two non-implementation limitations
+that stunt CageCoach and the broader goal of safe sharing
+oriented DFS:
+• No system can stop external data reconstruction.
+No matter if a user is using CageCoach, Google Drive,
+or any other hosting system, external actors with access to
+pieces of separate data can always reassemble it together.
+For example, an actor A has access to a subset of data 1, and
+an actor B has access to another subset of data 1. These two
+actors are not allowed access to either subset of data by the
+policies of the user whose data it is. However, this does not
+stop nor disincentivise actor A and B from simply sharing
+with each other the user’s data. Each filling in the other’s gap
+of missing data. No system can solve this if the requesting
+actors are able to observe data.
+• Leakage is still possible through indirect implicating
+fields.
+CageCoach’s redaction pipeline is quite rudimentary, in
+some cases data may be leaked through a combination of un-
+related fields. For example, with a street address, a malicious
+user may be able to correctly guess a zip code when paired
+with other information. This is due to CageCoach’s inability
+to understand the connections between data.
+CageCoach’s unique sharing-oriented DCFS structure pro-
+vides several new areas of exploration. CageCoach itself can
+be expanded to cover more datatypes, and work can be done
+to integrate the ingress of user’s data to the data hosts that
+CageCoach manages.
+6.1
+Collective Redaction Rules For
+Multi-Owner Data
+Given our system’s usage of a redaction pipeline, one could
+envision a scenario where data that is collected by one user,
+but contains multiple other users’ data is pass around each
+impacted user’s DCS for specific group based redaction. This
+would facilitate greater granularity of redaction and a sense
+of group ownership over data and its privacy implications.
+6.2
+Enhanced ACL And Redaction
+Through Impact Trees
+A future work could examine how to enhance the redaction
+rules to include field implications to provide greater coverage
+of privacy in the event a user misses these concepts them-
+selves. This would fill in the gaps that leaking implicating
+fields create.
+REFERENCES
+[1] Matt Blaze. 1993. A Cryptographic File System for UNIX. In Proceedings
+of the 1st ACM Conference on Computer and Communications Security
+(Fairfax, Virginia, USA) (CCS ’93). Association for Computing Machin-
+ery, New York, NY, USA, 9–16. https://doi.org/10.1145/168588.168590
+5
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+6
+

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf,len=363
+page_content='CageCoach: Sharing-Oriented Redaction-Capable  Distributed Cryptographic File System  Jason Carpenter  CARPE415@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='edu  University of Minnesota  Minneapolis, MN  Zhi-Li Zhang  zhzhang@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='edu  University of Minnesota  Minneapolis, MN  ABSTRACT  The modern data economy is built on sharing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' However,  sharing data can be an expensive and risky endeavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Exist-  ing sharing systems like Distributed File Systems provide full  read, write, and execute Role-based Access Control (RBAC)  for sharing data, but can be expensive and difficult to scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Likewise such systems operate on a binary access model for  their data, either a user can read all the data or read none of  the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This approach is not necessary for a more read-only  oriented data landscape, and one where data contains many  dimensions that represent a risk if overshared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' In order to  encourage users to share data and smooth out the process  of accessing such data a new approach is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This new  approach must simplify the RBAC of older DFS approaches  to something more read-only and something that integrates  redaction for user protections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  To accomplish this we present CageCoach, a simple sharing-  oriented Distributed Cryptographic File System (DCFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Cage-  Coach leverages the simplicity and speed of basic HTTP,  linked data concepts, and automatic redaction systems to  facilitate safe and easy sharing of user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The implemen-  tation of CageCoach is available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='edu/  CARPE415/CageCoach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  1  INTRODUCTION  User-generated data drives the modern world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Everything  from Uber driver rides and Google search queries to video  game experiences and Amazon purchase patterns feed user  data back into these systems to provide insights for improve-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Additionally, users sharing their data as part of crowd  sourcing solutions has proven key to reverse engineering gig  working applications such as Uber[5, 16, 18], Lyft[16, 18],  and Shipt[3, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Further these efforts help solve civic and  national needs such as with Atlanta’s Data Dashboard[13],  Minneapolis’s Opendata program[7], or the United State’s  Citizen Science initiative[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  However, users providing their data to these initiatives  often comes with a level of risk and a loss of control over the  data they provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Once a user has handed over information  the safety considerations, redaction approaches, and man-  agement decisions are out of their control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Further, should  any shared user data become dangerous to a user, the user  has no more sway to alleviate this risk other than ask the  current data holder to act, a practice often fruitless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  In order to further encourage users to share their data, a  new sharing oriented data hosting system is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Such a  platform must be simple to implement, easy to request data  from, but still provide some assurances of privacy and safety  for users involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Crucially it should remain in the user’s  control, and not be subject to control by others even those  hosting data such as on public hosting systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The privacy  capability must be granular not just in who can access data  but what specific data is accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' For example, for some  users, sharing their full name to everyone who asks is un-  reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Thus they should be able to share with some a  partial redaction of their name.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Existing works such as Dis-  tributed File Systems (DFS) are promising, but require exten-  sive implementation, Role-based Access Control (RBAC) en-  forcement, and do not implement granular redaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Other  platforms like Google Drive, Dropbox, and Kaggle are great  for sharing bulk data but also do not provide granular redac-  tion and require trusting of the platform holders to not share  otherwise redacted user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  In this work, we introduce CageCoach a sharing oriented  distributed cryptographic file system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' CageCoach’s notable  features are:  Simple Trustless DCFS built over HTTP GET/POST  Customizable RBAC and Datatype Granular Redac-  tion Pipeline  Easier sharing with Decentralized data access and  centralized user control  CageCoach streamlines the older RBAC based models of  DFSs and decentralizes the data hosting approaches of plat-  forms making for an overall simpler means of sharing data  with others while retaining granular privacy control for users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  This system is leverages simple HTTP GET/POST operations  to interact with symmetrically encrypted files hosted on any  HTTP platform to achieve decentralized hosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' These files  point back to their owners, represented by a controlling  server, that can facilitate redacted data access for a data re-  quester providing user control of data access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Finally, the  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='04214v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='CR]  10 Jan 2023  ,  user’s controlling server applies user defined redaction oper-  ations from a suite of modules CageCoach provides to reduce  sensitive data leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  CageCoach’s code can be found at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='edu/  CARPE415/CageCoach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  2  RELATED WORK  Distributed File Systems (DFS) and cryptographic file sys-  tems (DCFS) have been around for a long time with some  works as early as 1993[1] and as recent as 2020[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' These are  mature fields with well-developed and commercial products  we see every day, such as Dropbox, GoogleDrive, Hadoop,  Ceph, and others[22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Despite this, the changing data land-  scape and changing usage behaviors with data invite re-  examinations of existing systems to better fit them for a new  era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The work must relevant in the current data landscape,  data redaction, is an old field but with a renewed interest in  the face of big data breaches, data privacy concerns, and ma-  chine learning for data protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' In this section we outline  these two related areas and contrast them with our proposed  system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='1  Distributed & Cryptographic File  Systems  Distributed File Systems (DFS) are systems for maintaining  coherent file management across desperate hosting devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Examples include standard file hosting such as Google Drive,  Dropbox, and InRupt’s Solid[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Such systems have a long  history and continued relevance in the modern era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' DFS also  manifest as cloud storage systems, albeit with looser file sys-  tem format adherence to mesh with the more diverse Internet  access environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Extending DFSs into privacy and secu-  rity oriented spaces yields the Distributed Cryptographic  File System (DCFS) domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Works such as UPSS[2] focus  on creating a sharing-oriented and protective DFS with full  RBAC and mutable verifiable histories of each file involved as  a check against malicious behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Further other works such  as [10, 12] aim to utilize the blockchain to achieve the same  RBAC with a more decentralized approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Finally, other  approaches aim to refine key management in encryption for  DFS[14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='These systems while powerful, rely on relatively  expensive RBAC and infrastructure or require significant  trust for the platform holders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' In the former case, simplify-  ing the RBAC with the mostly read-only reality of user data  can lower RBAC complexity significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' In the latter case,  hosting infrastructure is still necessary, but one must create  a trustless environment in order to retain control of one’s  data even on such hosting platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Our work focuses on streamlining data sharing by creat-  ing a middlepoint between strong, rigid, and RBAC focused  approaches such as DCFSs and trust-oriented data platforms  and services like Uber, Kaggle, and Gridwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='2  Data Redaction  Data redaction is not a new field, but has gained vigor in the  last decade or so as the data economy has shaped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Redac-  tion provides the means for which sensitive data can be  made less sensitive and thus less dangerous in the event of  leaks, breaches, or theft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Likewise, redaction has its place in  academic publications when such publications may contain  in themselves dangerous or sensitive information[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Many  existing tools provide a user the quick means of redacting  a document such as [6] and [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' A handful of commercial  products, such as [21], [15], and [17], apply machine learning  to identify and remove automatically sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Finally,  other work such as [11] highlight an interesting scenario  where redaction itself must be transparent enough such that  the redaction doesn’t mislead the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' These systems  as implemented are not part of a sharing pipeline and are  applied ad-hoc to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' A system such as the one outlined by  UPSS[2], envisions such technologies are part of a pipeline  of data requests but did not implement or specify beyond  such designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Our work applies the concepts behind these redaction  systems, but crucially, as part of a standard granular access  pipeline and not as a one-off and static redaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This in  effect realizes some aspects of the UPSS[2] pipeline, but with-  out the more complex full RBAC suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  3  PROBLEM AND DESIGN GOALS  In order to build a system that encourages users to share their  data two primary problems and design considerations must  be achieved: Simplification of access control for accessing  and requesting data and automatic policy informed data  redaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' With these two aspects a sharing-oriented DFS  will lower the cost of sharing and accessing data and provide  a wide net of protections for users who choose to share.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='1  Simplify Access Control For Data  Existing DFS systems utilize a full suite of RBAC function-  ality to provide read, write, and execute functionality for  shared files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' These provisions while useful, require signif-  icant infrastructure such as certificates and user profiles  registered within the computational structure of the data  host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This full suite of RBAC is necessary if the group of  users intended to read, write, and/or execute the shared data,  but costly if sharing (read only) is the intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' By removing  the write and execute assumptions of RBAC we can in turn  simplify the operating infrastructure required for accessing  data and making sharing a lower cost effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This lower cost  is necessary for encouraging users to share their data, as it  2  CageCoach: Sharing-Oriented Redaction-Capable Distributed Cryptographic File System  ,  will be easier to host for consumption, and for consumers of  data as it will be easier to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='2  Provide Integrated Automatic User  Data Redaction  Regardless of ease of access, users must be given some as-  surances of safety, privacy, and proper use for their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Traditional RBAC focuses on binary access models for data,  either a user can read all the data or none of the data in a typi-  cally hosted file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This approach is not adequate for data items  that contain core sensitive fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' For example, a typical sales  receipt is useful for inventory systems and market trending  services, as they provide insights into purchases and sales  trends, however, these same receipts may contain the pur-  chaser’s name, credit card information, and/or address and  location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Such fields are not important for the overall trend,  but present a security risk for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' In a binary RBAC  model, such fields would available if the receipt is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  A more granular approach to access is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Such an ap-  proach is outlined but not realized or specified by UPSS[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Such an approach would require that when a user’s data is re-  quested by another, a trusted middle system acquires the raw  full set of data, and then redacts and removes information  that is included in the data but not allowed for that partic-  ular user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' For example, removing the name, address, and  credit fields from the sales receipt scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This approach  is required to provide granular and safer exposure of user’s  data for general consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Further, this process can be  handled by user-defined policy thus providing guidelines  for any user data added in the future thus lowering sharing  costs further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  4  CAGECOACH SYSTEM  We realize the goals of a sharing-oriented DFS with our  system CageCoach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' CageCoach simplifies the RBAC and  infrastructure of existing DFSs and integrates redaction tech-  nologies into a data request pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' All of this together  creates a simple and easy means for users to safely and eas-  ily share their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' CageCoach is organized around several  concepts and a flow, outlined in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Requesters, who re-  quest user data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Data hosts, which host encrypted data files  and some attached meta data files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Finally, a Data Control  Server (DCS) which manages the owner’s data, processes  requests made by requesters, and redacts outgoing sensitive  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' CageCoach’s operational use-case is:  (1) A owner uploads some data (video, text, audio, etc) to  a hosting system after encrypting and creating a meta  file for the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  (2) A requester sees this data and examines the meta file  (using GET for example) for information as to where  the owner’s DCS operates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Figure 1: CageCoach System, providing a streamlined  means for requestors to ask for data and receive useful  but protected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  (3) The requester sends a POST request to the owner’s  DCS server, asking to view the original data item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  (4) The DCS receives this request, verifies the requester’s  identity through asymmetric key phrase decryption,  and then uses GET to retrieve the remotely hosted  encrypted data file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  (5) The DCS decrypts the file with its own internal sym-  metric key and then applies a series of redaction oper-  ations on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  (6) The DCS forwards the remaining unredacted data to  the requester, completing the request and preventing  unnecessary or forbidden data from leaving encrypt-  ed/controlled space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  The details for how the RBAC is simplified and how the  redaction is integrated is detailed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='1  Simplifying RBAC Using HTTP And  Read-Only Assumptions  CageCoach simplifies the primary RBAC and infrastructure  of other DFSs by assuming that user data need only be read,  not written too or executed collaboratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Additionally,  unlike UPSS[2], since there is no write permissions data  versions are no longer necessary thus can relax the assump-  tion UPSS makes for needing a transparent modifications  tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' With this simplification in mind, CageCoach utilizes  the most common means of read-only operation on the Inter-  net: HTTP GET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This means that user data can be hosted on  any system that facilitates HTTP GET, such as open source  systems like Apache2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The data that gets hosted is the user’s  encrypted file and a plain text meta data file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Using some con-  cepts of linked data, the meta data file points to the owner’s  DCS to actually facilitate the request for data among other  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The total definition for this meta data file is:  3  (2)Directrequestertodataowner  B  (1)Requestaccesstodata  000  (3) Downloads encrypted file  Data Host  (Dropbox,GDrive,Apache)  000  000  HTTP Data Control  Requester  Server (Dcs)  AccessControl(ACL)And  DataCensoringRules(DCR)  (5)Alloweddatais returned  (4)Decryptsandprocessesfile,  owner-url: URL indicating where the owner’s DCS is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The  place where any request will be processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  meta-data: User filled info tags about the data, such as  what format it is, overall context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' All of this information is  optional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  description: A more textual description of the data, op-  tional if an owner wishes to provide more than just tags of  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  data-url: The URL indicating where the data this meta file  belongs to is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This is important for providing some backup if  the meta file is moved elsewhere or if it must live elsewhere  in hosting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  data-hash-sha1: A sha1 of the encrypted file to provide a  minimal check for any requester that wishes to double check  the file they are asking about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Despite our overall read-only approach, some computa-  tional efforts are still required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Namely the decryption of  the requested file and the granular redaction of information  within this file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The purpose of redirecting the requester  from the data host is to provide a centralized response by  the owner and the computational space for redaction poli-  cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The requester will send an HTTP POST request to the  DCS indicated by the owner-url and receive a decrypted  and redacted data file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The DCS’s process is implemented as  a basic python HTTP server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The process involves several  steps: 1) Receive a POST request with the URL of the data  being requested and optionally an ID and asymmetrically  encrypted phrase to verify the requester’s identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Cage-  Coach implements this with RSA public/private key pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  2) Locate the data profile for the requested data on the DCS  server, itself a simple text file containing pointers to decrypt  and identify the requested data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Additionally, if the user is  registered with the DCS (registry comprised of a private key  for decrypting phrases, the plain text passphrase, and a id  name) it will load their profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' We implement this as simply  a separate json file containing each requester’s information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Our approach assumes this registry happens outside of the  CageCoach architecture but can utilize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' 3) The DCS will  download the encrypted file from its host using HTTP GET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  After reception, the DCS will decrypt the data file and load  the redaction policies that match the specific data item (by  its name), the data type (json, mp3, etc), and finally the poli-  cies for the requester (if provided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' CageCoach implements  this encryption with symmetric keys using pythons Fernet  library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' 4) The DCS will apply these redaction operations,  gradually chipping away data until left with whatever is al-  lowed to pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' 5) The remaining data is sent to the requester  in the POST response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' The specifics of how the redaction is  applied is outlined in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Figure 2: CageCoach Redaction Pipeline, providing a  generalized measure of privacy assurance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='2  Access Control and Redaction  Pipelines  CageCoach’s read-only assumption for user data is not a  binary, like older models of RBAC based system, but granu-  lar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' By using a series of redaction operations over requested  data, CageCoach can allow partial access to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' These op-  erations, dividable by datatype as outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' 2, provide  for blurring faces in images, redacting text in jsons and csvs,  and muting specific words or background noises recognized  in audio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' In the overall data request pipeline after a user  has requested data and the DCS has downloaded the target  data, it will apply these redaction operations according to  the specific user, datatype, and data item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This provides three  levels of granularity for controlling data flow outwards to re-  questers: by datatype (all jsons, csvs, mp3s, etc), by data item  (ex: specific files like example-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='json hosted on Google Drive  or example-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='json hosted on dropbox), and by requester id  (ex: John Doe can access the user’s name, but Jane Doe can  only see the user’s first name).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' However, such operations  that would be specific to an owner, such as blurring only  the owner’s face, require the owner provide their own data  to the redacting DCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Our implementation we provide does  general redaction such as blurring all faces and removing a  handful of well known text fields such as social security and  street addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' We do not implement an audio redaction  approach as there isn’t a general python capable pre-built  audio redaction library nor a common set of what "words"  should be auto removed, unlike faces in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' CageCoach  does support extensions to these operations to tailor to spe-  cific users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Our implementation uses the Haar cascade and  OpenCV2 [9] python libraries for blurring faces (illustrated  with the blurring of photo of American Union Army General  Benjamin Butler fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' 3), and python Pandas to redact textual  data (example of such in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  4  010110  RedactionProcessesByDataTypes  010001  111011  Blob  Age:43  Name:"JohnDoe"  SSN:"999-99-9999"  File  CensorFields  Encrypted  8871  File  Image  CensorFaces/Persons  4)  Audio  CensorAudio SegmentsCageCoach: Sharing-Oriented Redaction-Capable Distributed Cryptographic File System  ,  Figure 3: CageCoach Redaction Pipeline example blur-  ring a specific image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Figure 4: CageCoach Redaction Pipeline example  redacting specific text and fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  5  CONCLUSION  In this work, we introduced a new sharing oriented imple-  mentation of DCFS: CageCoach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' CageCoach streamlines the  older RBAC heavy and trust-necessary hosting models of  DFS, while using the simpler HTTP GET/POST ecosystem to  facilitate easier data sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' All of this is possible while still  respecting the privacy of users through granular customize-  able redaction pipelines that handle removal of sensitive user  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  6  LIMITATIONS AND FUTURE WORK  CageCoach has a set of drawbacks and limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Cage-  Coach is implemented as a demonstration of a new inter-  pretation of sharing-oriented DCFS and not intended for  industrial or commercial use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Future implementations would  need to provide better integration with hosting services like  Google and Dropbox, and provide tougher and more robust  security checks and infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Likewise future work  improvements would be needed to make the redaction oper-  ations more capable and workable on a wider set of diverse  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Notably there are two non-implementation limitations  that stunt CageCoach and the broader goal of safe sharing  oriented DFS:  No system can stop external data reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  No matter if a user is using CageCoach, Google Drive,  or any other hosting system, external actors with access to  pieces of separate data can always reassemble it together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  For example, an actor A has access to a subset of data 1, and  an actor B has access to another subset of data 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' These two  actors are not allowed access to either subset of data by the  policies of the user whose data it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' However, this does not  stop nor disincentivise actor A and B from simply sharing  with each other the user’s data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Each filling in the other’s gap  of missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' No system can solve this if the requesting  actors are able to observe data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  Leakage is still possible through indirect implicating  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  CageCoach’s redaction pipeline is quite rudimentary, in  some cases data may be leaked through a combination of un-  related fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' For example, with a street address, a malicious  user may be able to correctly guess a zip code when paired  with other information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This is due to CageCoach’s inability  to understand the connections between data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  CageCoach’s unique sharing-oriented DCFS structure pro-  vides several new areas of exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' CageCoach itself can  be expanded to cover more datatypes, and work can be done  to integrate the ingress of user’s data to the data hosts that  CageCoach manages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='1  Collective Redaction Rules For  Multi-Owner Data  Given our system’s usage of a redaction pipeline, one could  envision a scenario where data that is collected by one user,  but contains multiple other users’ data is pass around each  impacted user’s DCS for specific group based redaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This  would facilitate greater granularity of redaction and a sense  of group ownership over data and its privacy implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content='2  Enhanced ACL And Redaction  Through Impact Trees  A future work could examine how to enhance the redaction  rules to include field implications to provide greater coverage  of privacy in the event a user misses these concepts them-  selves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' This would fill in the gaps that leaking implicating  fields create.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
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+page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
+page_content=' Citizen Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
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+page_content='  citizenscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
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+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE2T4oBgHgl3EQf8Ag-/content/2301.04214v1.pdf'}
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+arXiv:2301.01026v1  [cs.LG]  3 Jan 2023
+Continual Treatment Effect Estimation: Challenges and Opportunities
+Zhixuan Chu1, Sheng Li2
+1Ant Group, Hangzhou, China
+2University of Virginia, Charlottesville, USA
+chuzhixuan.czx@alibaba-inc.com, shengli@virginia.edu
+Introduction
+A further understanding of cause and effect within obser-
+vational data is critical across many domains, such as eco-
+nomics, health care, public policy, web mining, online ad-
+vertising, and marketing campaigns. Although significant
+advances have been made to overcome the challenges in
+causal effect estimation with observational data, such as
+missing counterfactual outcomes and selection bias between
+treatment and control groups, the existing methods mainly
+focus on source-specific and stationary observational data.
+Such learning strategies assume that all observational data
+are already available during the training phase and from only
+one source.
+Along with the fast-growing segments of industrial appli-
+cations, this assumption is unsubstantial in practice. Taking
+Alipay as an example, which is one of the world’s largest
+mobile payment platforms and offers financial services to
+billion-scale users, a tremendous amount of data containing
+much privacy-related information is produced daily and col-
+lected from different sources. In conclusion, the following
+two points are summed up. The first one is based on the
+characteristics of observational data, which are incremen-
+tally available from non-stationary data distributions. For
+instance, the electronic financial records for one marketing
+campaign are growing every day and they may be collected
+from different cities or even other countries. This character-
+istic implies that one cannot have access to all observational
+data at a one-time point and from one single source. The sec-
+ond reason is based on the realistic consideration of accessi-
+bility. For example, when new observational data are avail-
+able, if we want to refine the model previously trained by
+original data, maybe the original training data are no longer
+accessible due to a variety of reasons, e.g., legacy data may
+be unrecorded, proprietary, the sensitivity of financial data,
+too large to store, or subject to privacy constraint of personal
+information (Zhang et al. 2020). This practical concern of
+accessibility is ubiquitous in various academic and indus-
+trial applications. That’s what it boiled down to: in the era
+of big data, we face new challenges in causal inference with
+observational data, i.e., the extensibility for incrementally
+available observational data, the adaptability for extra do-
+Copyright © 2023, Association for the Advancement of Artificial
+Intelligence (www.aaai.org). All rights reserved.
+main adaptation problem except for the imbalance between
+treatment and control groups, and the accessibility for an
+enormous amount of data.
+In this position paper, we formally define the problem of
+continual treatment effect estimation, describe its research
+challenges, and then present possible solutions to this prob-
+lem. Moreover, we will discuss future research directions on
+this topic.
+Related Work
+Instead of randomized controlled trials, observational data
+is obtained by the researcher simply observing the subjects
+without any interference. That means that the researchers
+have no control over the treatment assignments, and they just
+observe the subjects and record data based on their obser-
+vations (Yao et al. 2021). Therefore, from the observational
+data, directly estimating the treatment effect is challenging
+due to the missing counterfactual outcomes and the exis-
+tence of confounders. Recently, powerful machine learning
+methods such as tree-based methods (Athey and Imbens
+2016; Wager and Athey 2018), representation learning
+(Li and Fu
+2017;
+Shalit, Johansson, and Sontag
+2017;
+Yao et al. 2018; Chu, Rathbun, and Li 2022), meta-learning
+(K¨unzel et al. 2019; Nie and Wager 2021), generative mod-
+els (Louizos et al. 2017; Yoon, Jordon, and van der Schaar
+2018) have achieved prominent progress in treatment effect
+estimation task.
+In addition, the combination of causal inference and
+other research fields also exhibits complementary strengths,
+such as computer vision (Tang et al. 2020; Liu et al. 2022a),
+graph learning (Ma et al. 2022; Chu, Rathbun, and Li 2021),
+natural language processing (Feder et al. 2022; Liu et al.
+2022b), and so on. The involved causal analysis helps to im-
+prove the model’s capability of discovering and resolving
+the underlying system beyond the statistical relationships
+learned from observational data.
+Problem Definition
+Suppose that the observational data contain n units collected
+from d different domains and the d-th dataset Dd contains
+the data {(x, y, t)|x ∈ X, y ∈ Y, t ∈ T } collected from d-th
+domain, which contains nd units. Let X denote all observed
+variables, Y denote the outcomes in the observational data,
+
+and T be a binary variable. Let D1:d = {D1, D2, ..., Dd}
+be the set of combination of d datasets, separately collected
+from d different domains. For d datasets {D1, D2, ..., Dd},
+they have the commonly observed variables, but due to the
+fact that they are collected from different domains, they have
+different distributions with respect to X, Y , and T in each
+dataset. Each unit in the observational data received one of
+two treatments. Let ti denote the treatment assignment for
+unit i; i = 1, ..., n. For binary treatments, ti = 1 is for
+the treatment group and ti = 0 for the control group. The
+outcome for unit i is denoted by yi
+t when treatment t is ap-
+plied to unit i. For observational data, only one of the poten-
+tial outcomes is observed. The observed outcome is called
+the factual outcome, and the remaining unobserved poten-
+tial outcomes are called counterfactual outcomes.
+This task can follow the potential outcome frame-
+work for estimating treatment effects
+(Rubin 1974;
+Splawa-Neyman, Dabrowska, and Speed 1990). The indi-
+vidual treatment effect (ITE) for unit i is the difference be-
+tween the potential treated and control outcomes and is de-
+fined as
+ITEi = yi
+1 − yi
+0.
+(1)
+The average treatment effect (ATE) is the difference be-
+tween the mean potential treated and control outcomes,
+which is defined as
+ATE = 1
+n
+n
+�
+i=1
+(yi
+1 − yi
+0).
+(2)
+The success of the potential outcome framework is based
+on the following assumptions (Imbens and Rubin 2015),
+which ensure that the treatment effect can be identified.
+Assumption 1 Stable Unit Treatment Value Assumption
+(SUTVA): The potential outcomes for any unit do not vary
+with the treatments assigned to other units, and, for each
+unit, there are no different forms or versions of each treat-
+ment level, which lead to different potential outcomes.
+Assumption 2 Consistency: The potential outcome of treat-
+ment t is equal to the observed outcome if the actual treat-
+ment received is t.
+Assumption 3 Positivity: For any value of x, treatment as-
+signment is not deterministic, i.e.,P(T = t|X = x) > 0, for
+all t and x.
+Assumption 4 Ignorability: Given covariates, treatment
+assignment is independent of the potential outcomes, i.e.,
+(y1, y0) ⊥⊥ t|x.
+Our goal is to develop a novel continual causal inference
+framework to estimate the causal effect for all available data,
+including new data Dd and the previous data D1:(d−1), with-
+out having access to previous data D1:(d−1).
+Research Challenges
+Existing causal effect inference methods, however, are un-
+able to deal with the aforementioned new challenges, i.e.,
+extensibility, adaptability, and accessibility. Although it is
+possible to adapt existing causal inference methods to cater
+to these issues, these adjusted methods still have inevitable
+defects. Three straightforward adaptation strategies are de-
+scribed as follows:
+1. If we directly apply the model previously trained based
+on original data to new observational data, the perfor-
+mance on new tasks will be very poor due to the domain
+shift issues among different data sources;
+2. Suppose we utilize newly available data to re-train the
+previously learned model for adapting changes in the
+data distribution. In that case, old knowledge will be
+completely or partially overwritten by the new one,
+which can result in severe performance degradation on
+old tasks. This is the well-known catastrophic forgetting
+problem (McCloskey and Cohen 1989; French 1999);
+3. To overcome the catastrophic forgetting problem, we
+may rely on the storage of old data and combine the old
+and new data together, and then re-train the model from
+scratch. However, this strategy is memory inefficient and
+time-consuming, and it brings practical concerns such as
+copyright or privacy issues when storing data for a long
+time (Samet, Miri, and Granger 2013).
+Any of these three strategies, in combination with the exist-
+ing causal effect inference methods, is deficient.
+Potential Solution
+To address the continual treatment effect estimation prob-
+lem, we propose a Continual Causal Effect Representation
+Learning framework (CERL) for estimating causal effect
+with incrementally available observational data. Instead of
+having access to all previous observational data, we only
+store a limited subset of feature representations learned from
+previous data. Combining selective and balanced represen-
+tation learning, feature representation distillation, and fea-
+ture transformation, our framework preserves the knowl-
+edge learned from previous data and updates the knowledge
+by leveraging new data so that it can achieve the continual
+causal effect estimation for incrementally new data without
+compromising the estimation capability for previous data.
+Framework Overview. To estimate the incrementally
+available observational data, the framework of CERL is
+mainly composed of two components: (1) the baseline
+causal effect learning model is only for the first available
+observational data, and thus we don’t need to consider the
+domain shift issue among different data sources. This com-
+ponent is equivalent to the traditional causal effect estima-
+tion problem; (2) the continual causal effect learning model
+is for the sequentially available observational data, where
+we need to handle more complex issues, such as knowledge
+transfer, catastrophic forgetting, global representation bal-
+ance, and memory constraint.
+Baseline Causal Effect Learning Model. We first train
+the baseline causal effect learning model for the initial obser-
+vational dataset and then bring in subsequent datasets. The
+task on the initial dataset can be converted to a traditional
+causal effect estimation problem. Owing to the success
+of deep learning for counterfactual inference, we propose
+
+to learn the selective and balanced feature representations
+(Shalit, Johansson, and Sontag 2017; Chu, Rathbun, and Li
+2020) for units in treatment and control groups and then in-
+fer the potential outcomes based on learned representation
+space.
+Sustainability of Model Learning. We have built the
+baseline model for causal effect estimation with observa-
+tional data from a single source. To avoid catastrophic for-
+getting when learning new data, we propose to preserve a
+subset of lower-dimensional feature representations rather
+than all original covariates. We also can adjust the number
+of preserved feature representations according to the mem-
+ory constraint.
+Continual Causal Effect Learning. We have stored mem-
+ory and the baseline model. To continually estimate the
+causal effect for incrementally available observational data,
+we incorporate feature representation distillation and feature
+representation transformation (Chu et al. 2023) to estimate
+the causal effect for all seen data based on a balanced global
+feature representation space.
+Research Opportunities
+Although significant advances have been made to over-
+come the challenges in causal effect estimation from an aca-
+demic perspective, industrial applications based on obser-
+vational data are always more complicated and harder. Un-
+like source-specific and stationary observational data, most
+real-world data are incrementally available and from non-
+stationary data distributions. Significantly, we also face the
+realistic consideration of accessibility. This work is the first
+attempt to investigate the continual lifelong causal effect in-
+ference problem and propose the corresponding evaluation
+criteria. However, constructing the comprehensive analyt-
+ical tools and the theoretical framework derived from this
+brand-new problem requires non-trivial efforts. Specifically,
+there are several potential directions for continual causal in-
+ference:
+• In addition to the distribution shift of the covariates
+among different domains, there are other potential tech-
+nical issues for continual effect estimation: for example,
+perhaps we do not initially observe all the necessary con-
+founding variables and may get access to increasingly
+more confounders.
+• Compared with homogeneous treatment effects (the
+magnitude and direction of the treatment effect are the
+same for all patients, regardless of any other patient char-
+acteristics), heterogeneous causal effects could differ for
+different individuals. This could be another candidate
+to consider for the continual treatment effect estimation
+model.
+• The basic assumptions for traditional causal effect esti-
+mation may not be completely applicable. New assump-
+tions may be supplemented, or previous assumptions
+need to be relaxed.
+• There exists a natural connection with continual domain
+adaptation among different times or domains (“contin-
+ual” causal inference) and between treatment and control
+groups (continual “causal inference”).
+• Compared to traditional causal effect estimation tasks
+based on a small amount of medical data, the continual
+causal inference method will face big data computing or
+cloud computing due to its objective task.
+• With the increasing public concern over privacy leakage
+in data, federated learning, which collaboratively trains
+the machine learning model without directly sharing the
+raw data among the data holders, may become a potential
+solution for continual causal inference.
+References
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+

B9AzT4oBgHgl3EQfGPvM/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf,len=419
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='01026v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='LG]  3 Jan 2023  Continual Treatment Effect Estimation: Challenges and Opportunities  Zhixuan Chu1, Sheng Li2  1Ant Group, Hangzhou, China  2University of Virginia, Charlottesville, USA  chuzhixuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='czx@alibaba-inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='com, shengli@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='edu  Introduction  A further understanding of cause and effect within obser-  vational data is critical across many domains, such as eco-  nomics, health care, public policy, web mining, online ad-  vertising, and marketing campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Although significant  advances have been made to overcome the challenges in  causal effect estimation with observational data, such as  missing counterfactual outcomes and selection bias between  treatment and control groups, the existing methods mainly  focus on source-specific and stationary observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Such learning strategies assume that all observational data  are already available during the training phase and from only  one source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Along with the fast-growing segments of industrial appli-  cations, this assumption is unsubstantial in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Taking  Alipay as an example, which is one of the world’s largest  mobile payment platforms and offers financial services to  billion-scale users, a tremendous amount of data containing  much privacy-related information is produced daily and col-  lected from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' In conclusion, the following  two points are summed up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' The first one is based on the  characteristics of observational data, which are incremen-  tally available from non-stationary data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' For  instance, the electronic financial records for one marketing  campaign are growing every day and they may be collected  from different cities or even other countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' This character-  istic implies that one cannot have access to all observational  data at a one-time point and from one single source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' The sec-  ond reason is based on the realistic consideration of accessi-  bility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' For example, when new observational data are avail-  able, if we want to refine the model previously trained by  original data, maybe the original training data are no longer  accessible due to a variety of reasons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=', legacy data may  be unrecorded, proprietary, the sensitivity of financial data,  too large to store, or subject to privacy constraint of personal  information (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' This practical concern of  accessibility is ubiquitous in various academic and indus-  trial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' That’s what it boiled down to: in the era  of big data, we face new challenges in causal inference with  observational data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=', the extensibility for incrementally  available observational data, the adaptability for extra do-  Copyright © 2023, Association for the Advancement of Artificial  Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  main adaptation problem except for the imbalance between  treatment and control groups, and the accessibility for an  enormous amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  In this position paper, we formally define the problem of  continual treatment effect estimation, describe its research  challenges, and then present possible solutions to this prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Moreover, we will discuss future research directions on  this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Related Work  Instead of randomized controlled trials, observational data  is obtained by the researcher simply observing the subjects  without any interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' That means that the researchers  have no control over the treatment assignments, and they just  observe the subjects and record data based on their obser-  vations (Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Therefore, from the observational  data, directly estimating the treatment effect is challenging  due to the missing counterfactual outcomes and the exis-  tence of confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Recently, powerful machine learning  methods such as tree-based methods (Athey and Imbens  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Wager and Athey 2018), representation learning  (Li and Fu  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Shalit, Johansson, and Sontag  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Yao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Chu, Rathbun, and Li 2022), meta-learning  (K¨unzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Nie and Wager 2021), generative mod-  els (Louizos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Yoon, Jordon, and van der Schaar  2018) have achieved prominent progress in treatment effect  estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  In addition, the combination of causal inference and  other research fields also exhibits complementary strengths,  such as computer vision (Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2022a),  graph learning (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Chu, Rathbun, and Li 2021),  natural language processing (Feder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  2022b), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' The involved causal analysis helps to im-  prove the model’s capability of discovering and resolving  the underlying system beyond the statistical relationships  learned from observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Problem Definition  Suppose that the observational data contain n units collected  from d different domains and the d-th dataset Dd contains  the data {(x, y, t)|x ∈ X, y ∈ Y, t ∈ T } collected from d-th  domain, which contains nd units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Let X denote all observed  variables, Y denote the outcomes in the observational data,  and T be a binary variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Let D1:d = {D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=', Dd}  be the set of combination of d datasets, separately collected  from d different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' For d datasets {D1, D2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=', Dd},  they have the commonly observed variables, but due to the  fact that they are collected from different domains, they have  different distributions with respect to X, Y , and T in each  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Each unit in the observational data received one of  two treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Let ti denote the treatment assignment for  unit i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' For binary treatments, ti = 1 is for  the treatment group and ti = 0 for the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' The  outcome for unit i is denoted by yi  t when treatment t is ap-  plied to unit i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' For observational data, only one of the poten-  tial outcomes is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' The observed outcome is called  the factual outcome, and the remaining unobserved poten-  tial outcomes are called counterfactual outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  This task can follow the potential outcome frame-  work for estimating treatment effects  (Rubin 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Splawa-Neyman, Dabrowska, and Speed 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' The indi-  vidual treatment effect (ITE) for unit i is the difference be-  tween the potential treated and control outcomes and is de-  fined as  ITEi = yi  1 − yi  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  (1)  The average treatment effect (ATE) is the difference be-  tween the mean potential treated and control outcomes,  which is defined as  ATE = 1  n  n  �  i=1  (yi  1 − yi  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  (2)  The success of the potential outcome framework is based  on the following assumptions (Imbens and Rubin 2015),  which ensure that the treatment effect can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Assumption 1 Stable Unit Treatment Value Assumption  (SUTVA): The potential outcomes for any unit do not vary  with the treatments assigned to other units, and, for each  unit, there are no different forms or versions of each treat-  ment level, which lead to different potential outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Assumption 2 Consistency: The potential outcome of treat-  ment t is equal to the observed outcome if the actual treat-  ment received is t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Assumption 3 Positivity: For any value of x, treatment as-  signment is not deterministic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=',P(T = t|X = x) > 0, for  all t and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Assumption 4 Ignorability: Given covariates, treatment  assignment is independent of the potential outcomes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=',  (y1, y0) ⊥⊥ t|x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Our goal is to develop a novel continual causal inference  framework to estimate the causal effect for all available data,  including new data Dd and the previous data D1:(d−1), with-  out having access to previous data D1:(d−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Research Challenges  Existing causal effect inference methods, however, are un-  able to deal with the aforementioned new challenges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=',  extensibility, adaptability, and accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Although it is  possible to adapt existing causal inference methods to cater  to these issues, these adjusted methods still have inevitable  defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Three straightforward adaptation strategies are de-  scribed as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' If we directly apply the model previously trained based  on original data to new observational data, the perfor-  mance on new tasks will be very poor due to the domain  shift issues among different data sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Suppose we utilize newly available data to re-train the  previously learned model for adapting changes in the  data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' In that case, old knowledge will be  completely or partially overwritten by the new one,  which can result in severe performance degradation on  old tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' This is the well-known catastrophic forgetting  problem (McCloskey and Cohen 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' French 1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' To overcome the catastrophic forgetting problem, we  may rely on the storage of old data and combine the old  and new data together, and then re-train the model from  scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' However, this strategy is memory inefficient and  time-consuming, and it brings practical concerns such as  copyright or privacy issues when storing data for a long  time (Samet, Miri, and Granger 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Any of these three strategies, in combination with the exist-  ing causal effect inference methods, is deficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Potential Solution  To address the continual treatment effect estimation prob-  lem, we propose a Continual Causal Effect Representation  Learning framework (CERL) for estimating causal effect  with incrementally available observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Instead of  having access to all previous observational data, we only  store a limited subset of feature representations learned from  previous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Combining selective and balanced represen-  tation learning, feature representation distillation, and fea-  ture transformation, our framework preserves the knowl-  edge learned from previous data and updates the knowledge  by leveraging new data so that it can achieve the continual  causal effect estimation for incrementally new data without  compromising the estimation capability for previous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Framework Overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' To estimate the incrementally  available observational data, the framework of CERL is  mainly composed of two components: (1) the baseline  causal effect learning model is only for the first available  observational data, and thus we don’t need to consider the  domain shift issue among different data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' This com-  ponent is equivalent to the traditional causal effect estima-  tion problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' (2) the continual causal effect learning model  is for the sequentially available observational data, where  we need to handle more complex issues, such as knowledge  transfer, catastrophic forgetting, global representation bal-  ance, and memory constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Baseline Causal Effect Learning Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' We first train  the baseline causal effect learning model for the initial obser-  vational dataset and then bring in subsequent datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' The  task on the initial dataset can be converted to a traditional  causal effect estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Owing to the success  of deep learning for counterfactual inference, we propose  to learn the selective and balanced feature representations  (Shalit, Johansson, and Sontag 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Chu, Rathbun, and Li  2020) for units in treatment and control groups and then in-  fer the potential outcomes based on learned representation  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Sustainability of Model Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' We have built the  baseline model for causal effect estimation with observa-  tional data from a single source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' To avoid catastrophic for-  getting when learning new data, we propose to preserve a  subset of lower-dimensional feature representations rather  than all original covariates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' We also can adjust the number  of preserved feature representations according to the mem-  ory constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Continual Causal Effect Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' We have stored mem-  ory and the baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' To continually estimate the  causal effect for incrementally available observational data,  we incorporate feature representation distillation and feature  representation transformation (Chu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2023) to estimate  the causal effect for all seen data based on a balanced global  feature representation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Research Opportunities  Although significant advances have been made to over-  come the challenges in causal effect estimation from an aca-  demic perspective, industrial applications based on obser-  vational data are always more complicated and harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Un-  like source-specific and stationary observational data, most  real-world data are incrementally available and from non-  stationary data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Significantly, we also face the  realistic consideration of accessibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' This work is the first  attempt to investigate the continual lifelong causal effect in-  ference problem and propose the corresponding evaluation  criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' However, constructing the comprehensive analyt-  ical tools and the theoretical framework derived from this  brand-new problem requires non-trivial efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Specifically,  there are several potential directions for continual causal in-  ference:  In addition to the distribution shift of the covariates  among different domains, there are other potential tech-  nical issues for continual effect estimation: for example,  perhaps we do not initially observe all the necessary con-  founding variables and may get access to increasingly  more confounders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Compared with homogeneous treatment effects (the  magnitude and direction of the treatment effect are the  same for all patients, regardless of any other patient char-  acteristics), heterogeneous causal effects could differ for  different individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' This could be another candidate  to consider for the continual treatment effect estimation  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  The basic assumptions for traditional causal effect esti-  mation may not be completely applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' New assump-  tions may be supplemented, or previous assumptions  need to be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  There exists a natural connection with continual domain  adaptation among different times or domains (“contin-  ual” causal inference) and between treatment and control  groups (continual “causal inference”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Compared to traditional causal effect estimation tasks  based on a small amount of medical data, the continual  causal inference method will face big data computing or  cloud computing due to its objective task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  With the increasing public concern over privacy leakage  in data, federated learning, which collaboratively trains  the machine learning model without directly sharing the  raw data among the data holders, may become a potential  solution for continual causal inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  References  Athey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
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+page_content=' ACM Transactions on Knowl-  edge Discovery from Data (TKDD), 15(5): 1–46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
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+page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
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+page_content=' Huai, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Gao, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
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+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Representation learning for treatment effect estimation from  observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Advances in Neural Information Pro-  cessing Systems, 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Yoon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Jordon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' and van der Schaar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' GANITE:  Estimation of individualized treatment effects using genera-  tive adversarial nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' In International Conference on Learn-  ing Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Ghosh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Li, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Tasci, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Heck, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='  Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' and Kuo, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' Class-incremental learn-  ing via deep model consolidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}
+page_content=' In The IEEE Winter Con-  ference on Applications of Computer Vision, 1131–1140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/B9AzT4oBgHgl3EQfGPvM/content/2301.01026v1.pdf'}

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