The 85-year-old, named locally as Eleanor Whitelaw, was found with life-threatening injuries following the attack at her home in Morningside on Friday afternoon.
A dedicated inquiry team has been set up by officers in the capital and extra patrols are being run in the streets around Morningside to reassure people.
It has been reported the pensioner, who lived alone, may have come across thieves who were robbing her property when she returned home.
However, police have also said people should beware of unsolicited callers and should take steps to keep themselves safe.
The advice issued yesterday stressed householders should keep doors locked and be wary of callers who arrive without an appointment.
People should always ask to see visitors' identification if they do not know them and call the police if they feel threatened.
Superintendent James Frame said: "I'd like to reassure the public that as well as a dedicated inquiry team investigating the incident, we have officers from Police Scotland carrying out patrols in the Morningside area to reassure residents and deal with any concerns local people may have.
"As always, we would also encourage residents to call us should they have concerns regarding anyone who calls on them unannounced in unusual circumstances or regarding anyone they perceive to be acting suspiciously around their home".
Local councillor Mark McInnes branded the attack despicable, adding: "The community will be clearly concerned that whoever was responsible for committing such a terrible crime is apprehended as soon as possible.
"I would urge anyone with any information they think might assist the police to contact them as soon as possible."
Officers cordoned off the street near Mrs Whitelaw's home at the weekend and were seen carrying out forensic tests inside after the investigation was launched.
Teams also searched through bushes at the side of the street and people's gardens and drains as they looked for clues to the identity of whoever was behind the attack on the pensioner.
The incident has shocked the community of Morningside, one of the capital's upmarket areas and home to many well-off pensioners.
Resident Eleanor Johnston, who lives in the flat opposite, said: "This is normally a very quiet area and residents are stunned that someone has been hurt so badly in her own home."
Another neighbour of Mrs Whitelaw's said: "I am totally shocked by this. It's mainly retired folk that live here and we're all talking about it because it's so frightening."
Detective Superintendent David McLaren, who said police are in the early stages of their investigation, have appealed for anyone in the area in the hours before the woman was found to get in contact.
"I would appeal to anyone who was in the Morningside Grove area between 2pm and 5pm on Friday to make contact with us as a matter of urgency," said Mr McLaren.
"I particularly want to hear from anyone who was in the area and saw anything unusual or anyone acting suspiciously."